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The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics.

189 Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY190 Polynomial invariants of finite groups, D. J. BENSON191 Finite geometry and combinatorics, F. DE CLERCK et al.192 Symplectic geometry, D. SALAMON (ed.)194 Independent random variables and rearrangement invariant spaces, M. BRAVERMAN195 Arithmetic of blowup algebras, W. VASCONCELOS196 Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND197 Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI et al.198 The algebraic characterization of geometric 4-manifolds, J. A. HILLMAN199 Invariant potential theory in the unit ball of C n , M. STOLL200 The Grothendieck theory of dessins d’enfant, L. SCHNEPS (ed.)201 Singularities, J.-P. BRASSELET (ed.)202 The technique of pseudodifferential operators, H. O. CORDES203 Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH204 Combinatorial and geometric group theory, A. J. DUNCAN, N. D. GILBERT & J. HOWIE (eds)205 Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds)207 Groups of Lie type and their geometries, W. M. KANTOR & L. DI MARTINO (eds)208 Vector bundles in algebraic geometry, N. J. HITCHIN, P. NEWSTEAD & W. M. OXBURY (eds)209 Arithmetic of diagonal hypersurfaces over infite fields, F. Q. GOUVEA & N. YUI210 Hilbert C∗-modules, E. C. LANCE211 Groups 93 Galway / St Andrews I, C. M. CAMPBELL et al. (eds)212 Groups 93 Galway / St Andrews II, C. M. CAMPBELL et al. (eds)214 Generalised Euler–Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO et al.215 Number theory 1992–93, S. DAVID (ed.)216 Stochastic partial differential equations, A. ETHERIDGE (ed.)217 Quadratic forms with applications to algebraic geometry and topology, A. PFISTER218 Surveys in combinatorics, 1995, P. ROWLINSON (ed.)220 Algebraic set theory, A. JOYAL & I. MOERDIJK221 Harmonic approximation, S. J. GARDINER222 Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds)223 Analytic semigroups and semilinear initial boundary value problems, KAZUAKI TAIRA224 Computability, enumerability, unsolvability, S. B. COOPER, T. A. SLAMAN & S. S. WAINER (eds)225 A mathematical introduction to string theory, S. ALBEVERIO et al.226 Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds)227 Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI & J. ROSENBERG (eds)228 Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds)229 Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J. W. S. CASSELS & E. V. FLYNN231 Semigroup theory and its applications, K. H. HOFMANN & M. W. MISLOVE (eds)232 The descriptive set theory of Polish group actions, H. BECKER & A. S. KECHRIS233 Finite fields and applications, S. COHEN & H. NIEDERREITER (eds)234 Introduction to subfactors, V. JONES & V. S. SUNDER235 Number theory 1993–94, S. DAVID (ed.)236 The James forest, H. FETTER & B. G. DE BUEN237 Sieve methods, exponential sums, and their applications in number theory, G. R. H. GREAVES et al.238 Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds)240 Stable groups, F. O. WAGNER241 Surveys in combinatorics, 1997, R. A. BAILEY (ed.)242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds)243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds)244 Model theory of groups and automorphism groups, D. EVANS (ed.)245 Geometry, combinatorial designs and related structures, J. W. P. HIRSCHFELD et al.246 p-Automorphisms of finite p-groups, E. I. KHUKHRO247 Analytic number theory, Y. MOTOHASHI (ed.)248 Tame topology and o-minimal structures, L. VAN DEN DRIES249 The atlas of finite groups: ten years on, R. CURTIS & R. WILSON (eds)250 Characters and blocks of finite groups, G. NAVARRO251 Grobner bases and applications, B. BUCHBERGER & F. WINKLER (eds)252 Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO, R. STOHR (eds)253 The q-Schur algebra, S. DONKIN254 Galois representations in arithmetic algebraic geometry, A. J. SCHOLL & R. L. TAYLOR (eds)255 Symmetries and integrability of difference equations, P. A. CLARKSON & F. W. NIJHOFF (eds)256 Aspects of Galois theory, H. VOLKLEIN et al.257 An introduction to noncommutative differential geometry and its physical applications 2ed, J. MADORE258 Sets and proofs, S. B. COOPER & J. TRUSS (eds)259 Models and computability, S. B. COOPER & J. TRUSS (eds)260 Groups St Andrews 1997 in Bath, I, C. M. CAMPBELL et al.261 Groups St Andrews 1997 in Bath, II, C. M. CAMPBELL et al.262 Analysis and logic, C. W. HENSON, J. IOVINO, A. S. KECHRIS & E. ODELL263 Singularity theory, B. BRUCE & D. MOND (eds)264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds)265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART267 Surveys in combinatorics, 1999, J. D. LAMB & D. A. PREECE (eds)268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJOSTRAND269 Ergodic theory and topological dynamics, M. B. BEKKA & M. MAYER270 Analysis on Lie Groups, N. T. VAROPOULOS & S. MUSTAPHA

271 Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV272 Character theory for the odd order theorem, T. PETERFALVI273 Spectral theory and geometry, E. B. DAVIES & Y. SAFAROV (eds)274 The Mandelbrot set, theme and variations, TAN LEI (ed.)275 Descriptive set theory and dynamical systems, M. FOREMAN et al.276 Singularities of plane curves, E. CASAS-ALVERO277 Computational and geometric aspects of modern algebra, M. D. ATKINSON et al.278 Global attractors in abstract parabolic problems, J. W. CHOLEWA & T. DLOTKO279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds)280 Characters and automorphism groups of compact Riemann surfaces, T. BREUER281 Explicit birational geometry of 3-folds, A. CORTI & M. REID (eds)282 Auslander–Buchweitz approximations of equivariant modules, M. HASHIMOTO283 Nonlinear elasticity, Y. FU & R. OGDEN (eds)284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SULI (eds)285 Rational points on curves over finite fields, H. NIEDERREITER & C. XING286 Clifford algebras and spinors 2ed, P. LOUNESTO287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE et al.288 Surveys in combinatorics, 2001, J. HIRSCHFELD (ed.)289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE290 Quantum groups and Lie theory, A. PRESSLEY (ed.)291 Tits buildings and the model theory of groups, K. TENT (ed.)292 A quantum groups primer, S. MAJID293 Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK294 Introduction to operator space theory, G. PISIER295 Geometry and Integrability, L. MASON & YAVUZ NUTKU (eds)296 Lectures on invariant theory, I. DOLGACHEV297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES298 Higher operands, higher categories, T. LEINSTER299 Kleinian Groups and Hyperbolic 3-Manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds)300 Introduction to Mobius Differential Geometry, U. HERTRICH-JEROMIN301 Stable Modules and the D(2)-Problem, F. E. A. JOHNSON302 Discrete and Continuous Nonlinear Schrodinger Systems, M. J. ABLOWITZ, B. PRINARI & A. D. TRUBATCH303 Number Theory and Algebraic Geometry, M. REID & A. SKOROBOOATOV (eds)304 Groups St Andrews 2001 in Oxford Vol. 1, C. M. CAMPBELL, E. F. ROBERTSON & G. C. SMITH (eds)305 Groups St Andrews 2001 in Oxford Vol. 2, C. M. CAMPBELL, E. F. ROBERTSON & G. C. SMITH (eds)306 Peyresq lectures on geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds)307 Surveys in Combinatorics 2003, C. D. WENSLEY (ed.)308 Topology, geometry and quantum field theory, U. L. TILLMANN (ed.)309 Corings and Comdules, T. BRZEZINSKI & R. WISBAUER310 Topics in Dynamics and Ergodic Theory, S. BEZUGLYI & S. KOLYADA (eds)311 Groups: topological, combinatorial and arithmetic aspects, T. W. MULLER (ed.)312 Foundations of Computational Mathematics, Minneapolis 2002, FELIPE CUCKER et al. (eds)313 Transcendantal aspects of algebraic cycles, S. MULLER-STACH & C. PETERS (eds)314 Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC315 Structured ring spectra, A. BAKER & B. RICHTER (eds)316 Linear Logic in Computer Science, T. EHRHARD et al. (eds)317 Advances in elliptic curve cryptography, I. F. BLAKE, G. SEROUSSI & N. SMART318 Perturbation of the boundary in boundary-value problems of Partial Differential Equations, DAN HENRY319 Double Affine Hecke Algebras, I. CHEREDNIK320 L-Functions and Galois Representations, D. BURNS, K. BUZZARD & J. NEKOVAR (eds)321 Surveys in Modern Mathematics, V. PRASOLOV & Y. ILYASHENKO (eds)322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI, N. C. SNAITH (eds)323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al. (eds)324 Singularities and Computer Algebra, C. LOSSEN & G. PFISTER (eds)325 Lectures on the Ricci Flow, P. TOPPING326 Modular Representations of Finite Groups of Lie Type, J. E. HUMPHREYS328 Fundamentals of Hyperbolic Manifolds, R. D. CANARY, A. MARDEN & D. B. A. EPSTEIN (eds)329 Spaces of Kleinian Groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds)330 Noncommutative Localization in Algebra and Topology, A. RANICKI (ed.)331 Foundations of Computational Mathematics, Santander 2005, L. PARDO, A. PINKUS, E. SULI & M. TODD (eds)332 Handbook of Tilting Theory, L. ANGELERI HUGEL, D. HAPPEL & H. KRAUSE (eds)333 Synthetic Differential Geometry 2ed, A. KOCK334 The Navier–Stokes Equations, P. G. DRAZIN & N. RILEY335 Lectures on the Combinatorics of Free Probability, A. NICA & R. SPEICHER336 Integral Closure of Ideals, Rings, and Modules, I. SWANSON & C. HUNEKE337 Methods in Banach Space Theory, J. M. F. CASTILLO & W. B. JOHNSON (eds)338 Surveys in Geometry and Number Theory, N. YOUNG (ed.)339 Groups St Andrews 2005 Vol. 1, C. M. CAMPBELL, M. R. QUICK, E. F. ROBERTSON & G. C. SMITH (eds)340 Groups St Andrews 2005 Vol. 2, C. M. CAMPBELL, M. R. QUICK, E. F. ROBERTSON & G. C. SMITH (eds)341 Ranks of Elliptic Curves and Random Matrix Theory, J. B. CONREY, D. W. FARMER, F. MEZZADRI & N. C.

SNAITH (eds)342 Elliptic Cohomology, H. R. MILLER & D. C. RAVENEL (eds)343 Algebraic Cycles and Motives Vol. 1, J. NAGEL & C. PETERS (eds)344 Algebraic Cycles and Motives Vol. 2, J. NAGEL & C. PETERS (eds)345 Algebraic and Analytic Geometry, A. NEEMAN346 Surveys in Combinatorics, 2007, A. HILTON & J. TALBOT (eds)347 Surveys in Contemporary Mathematics, N. YOUNG & Y. CHOI (eds)348 Transcendental Dynamics and Complex Analysis, P. RIPPON & G. STALLARD (eds)349 Model Theory with Applications to Algebra and Analysis Vol 1, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY

& A. WILKIE (eds)350 Model Theory with Applications to Algebra and Analysis Vol 2, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY

& A. WILKIE (eds)351 Finite von Neumann Algebras and Masas, A. SINCLAIR & R. SMITH352 Number Theory and Polynomials, J. MCKEE & C. SMYTH (eds)

Groups and Analysis

The legacy of Hermann Weyl

Edited by

KATRIN TENT

Universitat Bielefeld

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Contents

Preface page vii

1 Roe GoodmanHarmonic analysis on compact symmetric spaces 1

2 Erik van den BanWeyl, eigenfunction expansions, symmetric spaces 24

3 W.N. Everitt and H. KalfWeyl’s work on singular Sturm–Liouville operators 63

4 Markus J. PflaumFrom Weyl quantization to modern algebraic index theory 84

5 A.M. Hansson and A. LaptevSharp spectral inequalities for the Heisenberg Laplacian 100

6 Ursula HamenstädtEquidistribution for quadratic differentials 116

7 Werner MüllerWeyl’s law in the theory of automorphic forms 133

8 Daniel W. StroockWeyl’s Lemma, one of many 164

9 Christopher DeningerAnalysis on foliated spaces and arithmetic geometry 174

10 R.E. Howe, E.-C. Tan and J.F. WillenbringReciprocity algebras and branching 191

11 Jens Carsten JantzenCharacter formulae from Hermann Weyl to the present 232

12 Richard M. WeissThe Classification of affine buildings 271

13 Peter RoquetteEmmy Noether and Hermann Weyl 285

v

Preface

This volume grew out of the conference in honour of Hermann Weyl thattook place in Bielefeld in September 2006.

Weyl was born in 1885 in Elmshorn, a small town near Hamburg. Hestudied mathematics in Göttingen and Munich, and obtained his doc-torate in Göttingen under the supervision of Hilbert. After taking ateaching post for a few years, he left Göttingen for Zürich to accepta Chair of Mathematics at the ETH Zürich, where he was a colleagueof Einstein just at the time when Einstein was working out the detailsof the theory of general relativity. Weyl left Zürich in 1930 to becomeHilbert’s successor at Göttingen, moving to the new Institute for Ad-vanced Study in Princeton, New Jersey after the Nazis took power in1933. He remained there until his retirement in 1951. Together with hiswife, he spent the rest of his life in Princeton and Zürich, where he diedin 1955.

The Collaborative Resarch Centre (SFB 701) Spectral Structures andTopological Methods in Mathematics has manifold connections with theareas of mathematics that were founded or influenced by Weyl’s work.These areas include geometric foundations of manifolds and physics,topological groups, Lie groups and representation theory, harmonic anal-ysis and analytic number theory as well as foundations of mathematics.

In 1913, Weyl published Die Idee der Riemannschen Fläche (‘TheConcept of a Riemann Surface’), giving a unified treatment of Riemannsurfaces.

He described the development of relativity theory in his Raum, Zeit,Materie (‘Space, Time, Matter’) from 1918, which reached a fourth edi-tion in 1922. In 1918, he introduced the concept of gauge and gave thefirst example of what is now known as a gauge theory.

From 1923 to 1938, Weyl developed the theory of compact groupsin terms of matrix representations and proved a fundamental characterformula for compact Lie groups. His book Classical Groups openednew directions in invariant theory. It covered symmetric groups, general

vii

viii Preface

linear groups, orthogonal groups, and symplectic groups, and results ontheir invariants and representations.

In The Continuum, Weyl developed the logic of classical analysis alongthe lines of Brouwer’s intuitionism. However, he later decided that thisradical constructivism puts too much of a restriction on his mathematicsand reconciled himself with the more formalistic ideas of Hilbert.

Weyl also showed how to use exponential sums in diophantine approx-imation, with his criterion for uniform distribution modulo one, whichwas a fundamental contribution to analytic number theory.

During the conference, his lasting influence on current mathematicsbecame evident through a series of impressive talks often connectingtheorems of Weyl with the most current results in dynamical systems,invariant theory, or partial differential equations. We are happy that somany speakers agreed to contribute to this volume.

The conference was funded by the Collaborative Research Center(SFB 701) ’Spectral structures and topological methods in mathematics’.We gratefully acknowledge support by the German Research Foundation(DFG). Thanks are also due to Philip Herrmann for editing this volume,and to Markus Rost and Ulf Rehmann.

Bielefeld, December 2007

List of speakers and talks

M. Broue (Paris) C. Deninger (Münster)Complex Reflection Groups as WeylGroups

Determinants in von Neumann al-gebras and the entropy of noncom-mutativ group actions

R. Goodman (Rutgers) U. Hamenstädt (Bonn)Harmonic Analysis on CompactSymmetric Spaces - the Legacy ofH.Weyl and E.Cartan

Mixing properties of the Teich-müller flow

G. Huisken (Potsdam) J.C. Jantzen (Aarhus)The concept of Mass in GeneralRelativity

Character formulae from Weyl tothe present

H. Kalf (München) A. Laptev (Stockholm)Weyl’s work on singular Sturm-Liouville operators

Lieb–Thirring Inequalities. Recentresults

T. Lyons (Oxford) A. Macintyre (London)Inverting the signature of a path-extensions of a theorem of Chen

Some model theory of Lie groups

W. Müller (Bonn) N. Nadirashvili (Marseille)Weyl’s law and the theory of auto-morphic forms

Rearrangements and eigenvalues

M. Pflaum (Frankfurt) P. Roquette (Heidelberg)From Weyl quantization and Weylasymptotics to modern index the-ory.

Hermann Weyl and Emmy Noether(Some observations from the corre-spondence Hasse–Noether and otherdocuments)

M. Rost (Bielefeld) D. Salamon (Zürich)On Galois cohomology, norm func-tions and cycles

Pseudoholomorphic curves in sym-plectic topology

T.A. Springer (Utrecht) D.W. Stroock (MIT)A short history of the theory ofWeyl groups

Weyl’s Lemma, the original ellipticregularity result

E. van den Ban (Utrecht) R.M. Weiss (Tufts)Weyl, eigenfunction expansions andanalysis on non-compact symmetricspaces

Affine Buildings

x

1Harmonic Analysis on Compact Symmetric

Spaces: the Legacy of Elie Cartan andHermann Weyl

Roe GoodmanDepartment of Mathematics

Rutgers, The State University of New Jersey

1 Introduction

In his lecture Relativity theory as a stimulus in mathematical research[Wey4], Hermann Weyl says that “Frobenius and Issai Schur’s spadeworkon finite and compact groups and Cartan’s early work on semi-simpleLie groups and their representations had nothing to do with it [relativitytheory]. But for myself I can say that the wish to understand what reallyis the mathematical substance behind the formal apparatus of relativitytheory led me to the study of representations and invariants of groups,and my experience in this regard is probably not unique.”

Weyl’s first encounter with Lie groups and representation theory asa tool to understand relativity theory occurred in connection with theHelmholtz-Lie space problem and the problem of decomposing the tensorproduct ⊗kCn under the mutually commuting actions of the generallinear group GL(n, C) (on each copy of Cn ) and the symmetric groupSk (in permuting the k copies of Cn ).1 He later described the tensordecomposition problem in general terms [Wey3] as “an epistemologicalprinciple basic for all theoretical science, that of projecting the actualupon the background of the possible.” Mathematically, the issue was tofind subspaces of tensor space that are invariant and irreducible underall transformations that commute with Sk . This had already been doneby Frobenius and Schur around 1900, but apparently Weyl first becameaware of these results in the early 1920’s. The subspaces in question,which are the ranges of minimal projections in the group algebra of Sk ,are exactly the irreducible (polynomial) representations of GL(n, C), andall irreducible representations arise this way for varying k by includingmultiplication by integral powers of det(g) in the action. It seems clear

1 see [Haw, §11.2-3]

1

2 Roe Goodman

from his correspondence with Schur at this time that these results wereWeyl’s starting point for his later work in representation theory andinvariant theory.

Near the end of his monumental paper on representations of semisim-ple Lie groups [Wey1, Kap. IV, §4], Weyl considers the problem ofconstructing all the irreducible representations of a simply-connectedsimple Lie group G such as SL(n, C). This had been done on a case-by-case basis by Cartan [Car1], starting with the defining representationsfor the classical groups (or the adjoint representation for the exceptionalgroups) and building up a general irreducible representation by formingtensor products. By contrast, Weyl, following the example of Frobe-nius for finite groups, says that “the correct starting point for build-ing representations does not lie in the adjoint group, but rather in theregular representation, which through its reduction yields in one blowall irreducible representations.” He introduces the infinite-dimensionalspace C(U) of all continuous functions on the compact real form U of G

(U = SU(n) when G = SL(n, C)) and the right translation representa-tion of U on C(U). He then obtains the irreducible representations of U

and their characters by using the eigenspaces of compact integral oper-ators given by left convolution with positive-definite functions in C(U),in analogy with the decomposition of tensor spaces for GL(n, C) usingelements of the group algebra of Sk . The details are spelled out in thefamous Peter–Weyl paper [Pe-We], which proves that the normalizedmatrix entries of the irreducible unitary representations of U furnish anorthonormal basis for L2(U), and that every continuous function on U

is a uniform limit of linear combinations of these matrix entries.In the introduction to [Car2], É. Cartan says that his paper was in-

spired by the paper of Peter and Weyl, but he points out that for acompact Lie group their use of integral equations “gives a transcendentalsolution to a problem of an algebraic nature” (namely, the completenessof the set of finite-dimensional irreducible representations of the group).Cartan’s goal is “to give an algebraic solution to a problem of a tran-scendental nature, more general than that treated by Weyl.” Namely,to find an explicit decomposition of the space of all L2 functions ona homogeneous space into an orthogonal direct sum of group-invariantirreducible subspaces.

Cartan’s paper [Car2] then stimulated Weyl [Wey2] to treat the sameproblem again and write “the systematic exposition by which I shouldlike to replace the two papers Peter–Weyl [Pe-We] and Cartan [Car2].”In his characteristic style of finding the core of a problem through gen-

Harmonic analysis on compact symmetric spaces 3

eralization, Weyl takes the finite-dimensional irreducible subspaces offunctions (which he calls the harmonic sets by analogy with the case ofspherical harmonics) on the compact homogeneous space X as his start-ing point.2 Using the invariant measure on the homogenous space, heconstructs integral operators that intertwine the representation of thecompact group U on C(X) with the left regular representation on C(U).

In this paper we approach the Weyl–Cartan results by way of alge-braic groups. The finite functions on a homogeneous space for a com-pact connected Lie group (that is, the functions whose translates spana finite-dimensional subspace) can be viewed as regular functions on thecomplexified group (a complex reductive algebraic group). Irreduciblesubspaces of functions under the action of the compact group correspondto irreducible subspaces of regular functions on the complex reductivegroup—this is Weyl’s unitarian trick. We describe the algebraic groupversion of the Peter–Weyl decomposition and geometric criterion forsimple spectrum of a homogeneous space (due to E. Vinberg and B.Kimelfeld). We present R. Richardson’s algebraic group version of theCartan embedding of a symmetric space, and the celebrated results ofCartan and S. Helgason concerning finite-dimensional spherical repre-sentations.

We then turn to more recent results of J.-L. Clerc [Cle] concerningthe complexified Iwasawa decomposition and zonal spherical functionson a compact symmetric space, and S. Gindikin’s construction ([Gin1],[Gin2], [Gin3]) of the horospherical Cauchy–Radon transform, whichshows that compact symmetric spaces have canonical dual objects thatare complex manifolds.

We make frequent citations to the extraordinary books of A. Borel[Bor] and T. Hawkins [Haw], which contain penetrating historical ac-counts of the contributions of Weyl and Cartan. Borel’s book also de-scribes the development of algebraic groups by C. Chevalley that is basicto our approach. For a survey of other developments in harmonic analy-sis on symmetric spaces from Cartan’s paper to the mid 1980’s see Hel-gason [Hel3]. Thanks go to the referee for pointing out some notationalinconsistencies and making suggestions for improving the organizationof this paper.

2 Weyl’s emphasis on function spaces, rather than the underlying homogeneousspace, is in the spirit of the recent development of quantum groups; his imme-diate purpose was to make his theory sufficiently general to include also J. vonNeumann’s theory of almost-periodic functions on groups, in which the functionsdetermine a compactification of the underlying group.

4 Roe Goodman

2 Algebraic Group Version of Peter–Weyl Theorem

2.1 Isotypic Decomposition of O[X]

The paper [Pe-We] of Peter and Weyl considers compact Lie groups U ;because the group is compact left convolution with a continuous func-tion is a compact operator. Hence such an operator, if self-adjoint, hasfinite-dimensional eigenspaces that are invariant under right translationby elements of U . The finiteness of the invariant measure on U alsoguarantees that every finite-dimensional representation of U carries aU -invariant positive-definite inner product, and hence is completely re-ducible (decomposes as the direct sum of irreducible representations).3

Turning from Weyl’s transcendental methods to the more algebraicand geometric viewpoint preferred by Cartan, we recall that a subgroupG ⊂ GL(n, C) is an algebraic group if it is the zero set of a collectionof polynomials in the matrix entries. The regular functions O[G] arethe restrictions to G of polynomials in matrix entries and det−1 . Inparticular, G is a complex Lie group and the regular functions on G

are holomorphic. A finite-dimensional complex representation (π, V )of G is rational if the matrix entries of the representation are regularfunctions on G. The group G is reductive if every rational representationis completely reducible.

Let g be a complex semisimple Lie algebra. From the work of Cartan,Weyl, and Chevalley, one knows the following:

(1) There is a simply-connected complex linear algebraic group G withLie algebra g.

(2) The finite-dimensional representations of g correspond to rationalrepresentations of G.

(3) There is a real form u of g and a simply-connected compact Liegroup U ⊂ G with Lie algebra u.

(4) The finite-dimensional unitary representations of U extend uniquelyto rational representations of G, and U -invariant subspaces cor-respond to G-invariant subspaces.4

(5) The irreducible rational representations of G are parameterized bythe positive cone in a lattice of rank l (Cartan’s theorem of thehighest weight).5

3 This is the Hurwitz “trick” (Kunstgriff) that Weyl learned from I. Schur; seeHawkins [Haw, §12.2].

4 This is Weyl’s unitary trick.5 The first algebraic proofs of this that did not use case-by-case considerations were

found by Chevalley and Harish-Chandra in 1948; see [Bor, Ch. VII, §3.6-7].


The highest weight construction is carried out as follows: Fix a Borelsubgroup B = HN+ of G (a maximal connected solvable subgroup).Here H ∼= (C×)l , with l = rank(G), is a maximal algebraic torus in G,and N+ is the unipotent radical of B associated with a set of positiveroots of H on g. Let B = HN− be the opposite Borel subgroup. We canalways arrange the embedding G ⊂ GL(n, C) so that H consists of thediagonal matrices in G, N+ consists of the upper-triangular unipotentmatrices in G, and N− consists of the lower-triangular unipotent matri-ces in G. Let h be the Lie algebra of H and Φ ⊂ h∗ the roots of h ong. Write P (Φ) ⊂ h∗ for the weight lattice of H and P++ ⊂ P (Φ) for thedominant weights, relative to the system of positive roots determined byN+ . For λ ∈ P (Φ) we denote by h → hλ the corresponding character ofH. It extends to a character of B by (hn)λ = hλ for h ∈ H and n ∈ N+ .

An irreducible rational representation (π,E) of G is then determined(up to equivalence) by its highest weight. The subspace EN +

of N+ -fixed vectors in E is one-dimensional, and H acts on it by a characterh → hλ where λ ∈ P++ . The subspace EN −

of N−-fixed vectors in E isalso one-dimensional, and H acts on it by the character h → h−λ∗ whereλ∗ = −w0 · λ. Here w0 is the element of the Weyl group of (g, h) thatinterchanges positive and negative roots.

For each λ ∈ P++ we fix a model (πλ ,Eλ ) for the irreducible rationalrepresentation with highest weight λ. Then (πλ∗ , E

∗λ ) is the contragre-

dient representation. Fix a highest weight vector eλ ∈ Eλ and a lowestweight vector fλ∗ ∈ E∗

λ , normalized so that

〈eλ , fλ∗〉 = 1.

Here we are using 〈v, v∗〉 to denote the tautological duality pairing be-tween a vector space and its dual (in particular, this pairing is complexlinear in both arguments). For dealing with matrix entries as regularfunctions on the complex algebraic group G this is more convenient thanusing a U -invariant inner product on Eλ and identifying E∗

λ with Eλ viaa conjugate-linear map.

Let X be an irreducible affine algebraic G space. Denote the regularfunctions on X by O[X]. There is a representation ρ of G on O[X]:

ρ(g)f(x) = f(g−1x) for f ∈ O[X] and g ∈ G.

Because the G-action is algebraic, Spanρ(G)f is a finite-dimensionalrational G-module for f ∈ O[X]. There is a tautological G-intertwiningmap

Eλ ⊗HomG (Eλ,O[X]) → O[X],

6 Roe Goodman

given by v ⊗ T → Tv. For λ ∈ P++ let

O[X]N+(λ) = f ∈ O[X] : ρ(hn)f = hλf for h ∈ H and n ∈ N+.

(2.1)The key point is that the choice of a highest weight vector eλ gives anisomorphism

HomG (Eλ,O[X]) ∼= O[X]N+(λ). (2.2)

Here a G-intertwining map T applied to the highest weight vector givesthe function ϕ = Teλ ∈ O[X]N

+(λ), and conversely every such function

ϕ defines a unique intertwining map T by this formula.6 From (2.2)we see that the highest weights of the G-irreducible subspaces of O[X]comprise the set

Spec(X) = λ ∈ P++ : O[X]N+(λ) = 0 (the G spectrum of X)

Using the isomorphism (2.2) and the reductivity of G, we obtain thedecomposition of O[X] under the action of G, as follows:

Theorem 2.1 The isotypic subspace of type (πλ ,Eλ ) in O[X] is thelinear span of the G-translates of O[X]N

+(λ). Furthermore,

O[X] ∼=⊕

λ∈Spec(X )

Eλ ⊗O[X]N+(λ) (algebraic direct sum) (2.3)

as a G-module, with action πλ (g)⊗ 1 on the λ summand.

The action of G on O[X] is not only linear; it also preserves thealgebra structure. Since O[X]N

+(λ) ·O[X]N

+(µ) ⊂ O[X]N

+(λ+µ) under

pointwise multiplication andO[X] has no zero divisors (X is irreducible),it follows from (2.3) that

Spec(X) is an additive subsemigroup of P++ .

The multiplicity of πλ in O[X] is dimO[X]N+(λ) (which may be infi-

nite). All of this was certainly known (perhaps in less precise form) byCartan and Weyl at the time [Pe-We] appeared. We now consider Car-tan’s goal in [Car2] to determine the decomposition (2.3) when G actstransitively on X; especially, when X is a symmetric space. This requiresdetermining the spectrum and the multiplicities in this decomposition.

6 Weyl uses a similar construction in [Wey2], defining intertwining maps by integra-tion over a compact homogeneous space.


2.2 Multiplicity Free Spaces

We say that an irreducible affine G-space X is multiplicity free if all theirreducible representations of G that occur in O[X] have multiplicityone. Thanks to the theorem of the highest weight, this property canbe translated into a geometric statement (see [Vi-Ki]). For a subgroupK ⊂ G and x ∈ X write Kx = k ∈ L : k · x = x for the isotropygroup at x.

Theorem 2.2 (Vinberg–Kimelfeld) Suppose there is a point x0 ∈ X

such that B · x0 is open in X. Then X is multiplicity free. In this case,if λ ∈ Spec(X) then hλ = 1 for all h ∈ Hx0 .

Proof If B · x0 is open in X, then it is Zariski dense in X (since X isirreducible). Hence f ∈ O[X]N

+(λ) is determined by f(x0), since on the

dense set B · x0 it satisfies f(b · x0) = b−λf(x0). In particular, if f = 0then f(x0) = 0, and hence hλ = 1 for all h ∈ Hx0 . Thus

dimO[X]N+(λ) ≤ 1 for all λ ∈ P++ .

Now apply Theorem 2.1.

Remark. The converse to Theorem 2.2 is true; this depends on someresults of Rosenlicht [Ros] and is the starting point for the classificationof multiplicity free spaces (see [Be-Ra]).

Example: Algebraic Peter–Weyl Decomposition

Theorem 2.2 implies the algebraic version of the Peter-Weyl decompo-sition of the regular representation of G. Consider the reductive groupG × G acting on X = G by left and right translations. Denote thisrepresentation by ρ:

ρ(y, z)f(x) = f(y−1xz), for f ∈ O[G] and x, y, z ∈ G.

Take H ×H as the Cartan subgroup and B ×B as the Borel subgroupof G × G. Let x0 = I (the identity in G). The orbit of x0 under theBorel subgroup is

(B ×B) · x0 = N−HN+ (Gauss decomposition) (2.4)

This orbit is open in G since g = n− + h + n+ . Hence G is multiplicityfree as a G×G space. The G×G highest weights (relative to this choiceof Borel subgroup) are pairs (w0µ, λ), with λ, µ ∈ P++ . The diagonal

8 Roe Goodman

subgroup H = (h, h) : h ∈ H fixes x0 , so if (w0µ, λ) occurs as ahighest weight in O[X], then

hw 0 µ+λ = 1 for all h ∈ H.

This means that µ = −w0λ = λ∗; hence Eµ = Eλ∗ is the contragredientrepresentation of G.

Now set ψλ (g) = 〈πλ (g)eλ , fλ∗〉. This function satisfies ψλ (x0) = 1and

ψλ(b−1gb) = 〈πλ (g)πλ (b)eλ , πλ∗(b)fλ∗〉 = bλ bw 0 λ∗ψλ(g)

for b ∈ B and b ∈ B. Hence ψλ is a B × B highest weight vector forG × G of weight (w0λ

∗, λ). This proves that Spec(X) = (w0λ∗, λ) :

λ ∈ P++.

Theorem 2.3 For λ ∈ P++ let Vλ = Spanρ(G × G)ψλ. Then Vλ∼=

Eλ∗ ⊗ Eλ as a G×G module. Furthermore,

O[G] =⊕

λ∈P+ +

Vλ . (2.5)

In particular, O[G] is multiplicity free as a G ×G module, while underthe action of G × 1 it decomposes into the sum of dim Eλ copies of Eλ

for all λ ∈ P++ .

The function ψλ in Theorem 2.3 is called the generating function [Žel]for the representation πλ . Since ψλ (n−hn+) = hλ and N−HN+ is densein G, it is clear that

ψλ(g)ψµ(g) = ψλ+µ(g). (2.6)

The semigroup P++ of dominant integral weights is free with generatorsλ1 , . . . , λl , called the fundamental weights.

Proposition 2.4 (Product Formula) Set ψi(g) = ψλi(g). Let λ ∈ P++

and write λ = m1λ1 + · · ·+ mlλl with mi ∈ N. Then

ψλ(g) = ψ1(g)m 1 · · ·ψl(g)ml for g ∈ G. (2.7)

Remark. From the product formula it is evident that the existenceof a rational representation with highest weight λ is equivalent to theproperty that the functions n−hn+ → hλi on N−HN+ extend to regularfunctions on G for i = 1, . . . , l.


Example. Suppose G = SL(n, C). Take B as the group of upper-triangular matrices. We may identify P with Zn , where λ = [λ1 , . . . , λn ]gives the character

hλ = xλ11 · · ·xλn

n , h = diag[x1 , . . . , xn ].

Then P++ consists of the monotone decreasing n-tuples and is generatedby

λi = [1, . . . , 1︸︷︷︸i

, 0, . . . , 0] for i = 1, . . . , n− 1.

The fundamental representations are the exterior powers Eλi=∧i Cn of

the defining representation, for i = 1, . . . , n−1. The generating functionψi(g) is the ith principal minor of g. The Gauss decomposition (2.4) isthe familiar LDU matrix factorization from linear algebra, and

N−HN+ = g ∈ SL(n, C) : ψi(g) = 0 for i = 1, . . . , n− 1 .

Let K ⊂ G be a subgroup and let O[G]R(K ) be the right K-invariantregular functions on G (those functions f such that f(gk) = f(g) for allk ∈ K). This subspace of O[G] is invariant under left translations by G.

Corollary 2.1 Let EKλ be the subspace of K-fixed vectors in Eλ . Then

O[G]R(K ) ∼=⊕

λ∈P+ +

Eλ ⊗ EKλ∗ (2.8)

as a G module under left translations, with G acting by πλ ⊗ 1 on the λ-isotypic summand. Thus the multiplicity of πλ in O[G]R(K ) is dim EK

λ∗ .

For any closed subgroup K of G whose Lie algebra is a complex sub-space of g, the coset space G/K is a complex manifold on which G actsholomorphically, and the elements of O[G]R(K ) are holomorphic func-tions on G/K. When K is a reductive algebraic subgroup, then themanifold G/K also has the structure of an affine algebraic G-space suchthat the regular functions are exactly the elements of O[G]R(K ) (a re-sult of Matsushima [Mat]; see also Borel and Harish-Chandra [Bo-Ha]).Also, when K is reductive then dim EK

λ∗ = dim EKλ , since the identity

representation is self-dual.The pair (G,K) is called spherical if

dim EKλ ≤ 1 for all λ ∈ P++ .

In this case, we refer to K as a spherical subgroup of G. When K is

10 Roe Goodman

reductive, this property is equivalent to G/K being a multiplicity-freeG-space, by Corollary 2.1.

3 Complexifications of Compact Symmetric Spaces

3.1 Algebraic Version of Cartan Embedding

Cartan’s paper [Car2] studies the decomposition of C(U/K0), where U isa compact real form of the simply-connected complex semisimple groupG and K0 = Uθ is the fixed-point set of an involutive automorphismθ of U . The compact symmetric space X = U/K0 is simply-connectedand hence the group K0 is connected.7 The involution extends uniquelyto an algebraic group automorphism of G that we continue to denoteas θ. The algebraic subgroup group K = Gθ is connected and is thecomplexification of K0 in G, hence reductive. By Matsushima’s theoremG/K is an affine algebraic variety. It can be embedded into G as an affinealgebraic subset as follows (see [Ric1], [Ric2]):

Define

g y = gyθ(g)−1 , for g, y ∈ G.

We have (g (h y)) = (gh) y for g, h, y ∈ G, so this gives an action ofG on itself which we will call the θ-twisted conjugation action. Let

Q = y ∈ G : θ(y) = y−1.

Then Q is an algebraic subset of G. Since θ(g y) = θ(g)y−1g−1 =(g y)−1 , we have G Q = Q.

Theorem 3.1 (Richardson) The θ-twisted action of G is transitive oneach irreducible component of Q. Hence Q is a finite union of Zariski-closed θ-twisted G-orbits.

The proof consists of showing that the tangent space to a twistedG-orbit coincides with the tangent space to Q.

Corollary 3.1 Let P = G 1 = gθ(g)−1 : g ∈ G be the orbit of theidentity element under the θ-twisted conjugation action. Then P is aZariski-closed irreducible subset of G isomorphic to G/K as an affineG-space (relative to the θ-twisted conjugation action of G).

7 This theorem of Cartan extends Weyl’s results for compact semisimple groups–seeBorel [Bor, Chap. IV, §2].


There is a θ-stable noncompact real form G0 of G so that K0 is amaximal compact subgroup of G0 . The symmetric space G0/K0 is thenoncompact dual to U/K0 . The Cartan embedding is the map G0/K0 →P0 ⊂ G0 , where P0 = G0 1 = exp p0 and p0 is the −1 eigenspace of θ

in g0 (P0 is Cartan’s space E–see Borel [Bor, Ch. IV, §2.4]).

3.2 Classical Examples

Let G ⊂ GL(n, C) be a connected classical group whose Lie algebra issimple. The involutions and associated symmetric spaces G/K for G canbe described in terms of the following three kinds of geometric structureson Cn (in the second and third type, G is the isometry group of the formand K is the subgroup preserving the indicated decomposition of Cn ):

(1) nondegenerate bilinear forms G = SL(n, C) and K = SO(n, C)or Sp(n, C)

(2) polarizations Cn = V+ ⊕ V− with V± totally isotropic subspacesfor a bilinear form (zero or nondegenerate)

(3) orthogonal decompositions Cn = V+ ⊕ V− with V± nondegen-erate subspaces for a nondegenerate bilinear form

The proof that these structures give all the possible involutive auto-morphisms of the classical groups (up to inner automorphisms) can beobtained from following characterization of automorphisms of the clas-sical groups:

Proposition 3.2 Let σ be a regular automorphism of the classical groupG.

(1) If G = SL(n, C) then there exists s ∈ G so that σ is either σ(g) =sgs−1 or σ(g) = s(gt)−1s−1 .

(2) If G is Sp(n, C) then there exists s ∈ G so that σ(g) = sgs−1 .(3) If G is SO(n, C) with n = 2, 4, then there exists s ∈ O(n, C) so that

σ(g) = sgs−1 .

Proof The Weyl dimension formula implies that the defining representa-tion (and its dual, in the case G = SL(n, C)) is the unique representationof smallest dimension. So this representation is sent to an equivalent rep-resentation (or its dual) by σ. The existence of the element s followsfrom this equivalence (see [Go-Wa, §11.2.4] for details).8

8 This type of result was one motivation for Weyl to learn Cartan’s theory of repre-sentations of semisimple Lie groups–see Borel[Bor, Chap. III, §1] for more details.

12 Roe Goodman

Example. Let G = SL(n, C) and θ(g) = (gt)−1 . Then

K = SO(n, C), U = SU(n), K0 = SO(n), G0 = SL(n, R).

Also g y = gygt and Q = y ∈ G : yt = y = P , so there is one orbit.Hence the map gK → ggt gives the algebraic embedding

SL(n, C)/SO(n, C) ∼= y ∈ Mn (C) : y = yt , det y = 1.

For the other classical examples, see Goodman–Wallach [Go-Wa, §11.2.5].

3.3 Complexified Iwasawa Decomposition

The real semisimple Lie algebra g0 has a Cartan decomposition g0 =k0 +p0 into +1 and −1 eigenspaces of the Cartan involution θ. The non-compact real group G0 has an Iwasawa decomposition 9 G0 = K0A0N0 .Here A0 = exp a0 is a vector group with a0 a maximal abelian subspaceof p0 , and N0 is a nilpotent subgroup normalized by A0 . Let A andN be the complexifications of A0 and N0 in G, respectively. Then A

is a complex algebraic torus of rank l (the rank of G/K) and N is aunipotent subgroup. There is a θ-stable Cartan subgroup H of G suchthat A ⊂ H and the following holds (see Vust [Vus] for the general caseand Goodman-Wallach [Go-Wa, §12.3.1] for the classical groups):

(1) KAN is a Zariski-dense subset of G.(2) The subgroup M = CentK (A) is reductive and normalizes N .(3) Let T = H ∩K. Then H = AT and A ∩ T is finite.(4) There exists a Borel subgroup B with HN ⊂ B ⊂MAN .

Thus MAN is a parabolic subgroup of G with reductive Levi componentMA and unipotent radical N . We will give a more precise descriptionof the set KAN in the next section.

4 Representations on Symmetric Spaces

4.1 Spherical Representations

We continue with the same setting and notation as in Section 3.3; inparticular, P++ is the set of B-dominant weights. If λ ∈ P++ andEK

λ = 0 then λ will be called a K spherical highest weight and Eλ a K

spherical representation.

9 When G0 = SL(n, R) this decomposition is the so-called QR factorization of amatrix obtained by the Gram-Schmidt orthogonalization algorithm.


Proposition 4.1

(i) K is a spherical subgroup of G.(ii) Let T = H ∩K. If λ ∈ P++ is a K spherical highest weight, then

tλ = 1 for all t ∈ T. (4.1)

Proof Since B contains AN , the Iwasawa decomposition shows that BK

is dense in G, so B has an open orbit on G/K. Hence K is a sphericalsubgroup by Theorem 2.2. Since T is the stabilizer in H of the pointK ∈ G/K, condition (4.1) likewise holds.

We say that λ is θ-admissible if it satisfies (4.1).

Example. Let G = SL(n,C) and θ(g) = (gt)−1 . Here A = H

(diagonal matrices in G), N = all upper-triangular unipotent matrices,and M = T ∼= (Z/2Z)n−1 consists of all matrices

t = diag[δ1 , . . . , δn ], δi = ±1, det(t) = 1.

Hence the θ-admissible highest weights λ = [λ1 , . . . , λn−1 , 0] are thosewith λi even for all i.

Remark. In general, the subgroup F = T ∩A is finite and consists ofelements of order 2, since h = θ(h) = h−1 for h ∈ F . Thus a θ-admissiblehighest weight λ is trivial on T and its restriction to A is even, in thesense that hλ = 1 for h ∈ F .

Cartan [Car2] proved the implication (i) =⇒ (iii) in the followingtheorem and gave some indications for the proof of the converse (seeBorel [Bor, Chap. IV §4.4-5]). Thus the following result is sometimescalled the Cartan–Helgason theorem, although part (ii) and the firstcomplete proof of the theorem is due to Helgason [Hel1].

Theorem 4.2 Let (πλ ,Eλ ) be an irreducible rational representation ofG with highest weight λ (relative to B). The following are equivalent:

(i) EKλ = 0.

(ii) MN fixes the B-highest weight vector in Eλ .(iii) λ is θ-admissible.

Proof The equivalence of (ii) and (iii) follows by a Lie algebra argumentusing sl2 representation theory (see [Hel1] or [Go-Wa, §12.3.3]), and the

14 Roe Goodman

implication (i) =⇒ (iii) comes from Proposition 4.1. We give Helgason’sanalytic proof that (iii) =⇒ (i).10 Let λ be θ-admissible. Define

v0 =∫

K 0

πλ(k)eλ dk. (4.2)

Then v0 ∈ (Eλ )K by the unitarian trick, since K is connected. To showv0 = 0, let ψλ be the generating function for πλ . Then

〈v0 , fλ∗〉 =∫

K 0

ψλ (k) dk. (4.3)

We use the following properties:

(1) Let σ be the complex conjugation of G whose fixed-point set is G0 .Then χ(σ(a)) = χ(a) for any regular character χ of A.

(2) If h ∈ H ∩G0 = (T ∩G0)A0 , then hλ > 0 by (1), since h = t exp(x)with t ∈ T and x ∈ a0 .

(3) ψλ (g) ≥ 0 for g ∈ G0 by (2) and the Gauss decomposition.

Since ψλ(1) = 1 and K0 ⊂ G0 , property (3) shows that the integral (4.3)is nonzero.

Example. Let G = SL(n, C) and θ(g) = (gt)−1 . Here A0 consists ofreal diagonal matrices, G0 = SL(n, R), and

ψλ(g) = det1(g)m 1 · · · detn−1(g)mn −1 ,

where deti is the ith principal minor and mi = λi − λi+1 . Since λ isθ-admissible iff all λi are even, condition (3) in the proof of Theorem 4.2obviously holds. For example, the highest weight λ = [2, 0, . . . , 0] is ad-missible, and the corresponding spherical representation Eλ = S2(Cn ).The K-fixed vector in Eλ is

∑i ei ⊗ ei , where ei is the standard basis

for Cn .

The l fundamental K-spherical highest weights µ1 , . . . , µr (with l =dim A the rank of G/K) are linearly independent, and the general spher-ical highest weight is µ = m1µ1 + · · ·+mlµl with mi ∈ N (see [Hel2, Ch.V, §4]). Let Λ ⊂ P++ be the subsemigroup of spherical highest weights.Since K is reductive and the identity representation is self-dual, EK

λ = 0if and only if EK

λ∗ = 0. Hence Λ is invariant under the map λ → λ∗ onP++ .

Corollary 4.1 As a G-module, O[G/K] ∼=⊕

µ∈Λ Eµ .

10 An algebraic-geometric proof was given later by Vust [Vus].


4.2 Zonal Spherical and Horospherical Functions

For each µ ∈ Λ choose a K-fixed spherical vector eKµ ∈ Eµ and a MN -

fixed conical vector eµ ∈ Eµ , normalized so that

〈eµ , eKµ∗〉 = 1, 〈eK

µ , eKµ∗〉 = 1. (4.4)

The zonal spherical function ϕµ ∈ O[G] is the representative functiondetermined by pairing the K-fixed vectors in Eµ and Eµ∗ :

ϕµ(g) = 〈πµ(g)eKµ , eK

µ∗〉.

From the definition it is clear that

ϕµ(kgk′) = ϕµ(g) and ϕµ(1) = 1

for k, k′ ∈ K and g ∈ G. Thus ϕµ is a regular function on G/K that isconstant on the K-orbits.

The zonal horospherical function ∆µ ∈ O[G] is the representativefunction determined by pairing the MN -fixed vector in Eµ with theK-fixed vector in Eµ∗ :

∆µ(g) = 〈πµ(g)eµ , eKµ∗〉.

From the definition it is clear that

∆µ(kgman) = aµ∆µ(g) and ∆(1) = 1 (4.5)

for k ∈ K, g ∈ G, and man ∈ MAN . Properties (4.5) with g = 1determine ∆µ uniquely, since KAN is dense in G. We can view ∆µ asa holomorphic function on the affine symmetric space K\G that trans-forms by the character man → aµ along the MAN orbits. The existenceof a regular function on G with these transformation properties is equiva-lent to the existence of the K-spherical representation πµ (just as for thegenerating functions ψλ in Section 2.2, which are the zonal horospheri-cal functions associated with the diagonal embedding of G as a sphericalsubgroup of G×G). Let µ and ν be K-spherical highest weights. From(4.5) and the density of the set KMAN it follows that

∆µ(g)∆ν (g) = ∆µ+ν (g) for g ∈ G. (4.6)

Let µ1 , . . . , µr be the fundamental K-spherical highest weights, anddefine11

∆j (g) = ∆µj (g).

11 Gindikin [Gin3] calls ∆j the Sylvester functions; Theorem 4.3 shows they playthe same role for the KAN decomposition as the generating functions ψj forthe N−HN + decomposition.

16 Roe Goodman

For a general K-spherical highest weight µ = m1µ1 + · · ·+mrµr formula(4.6) implies the product formula

∆µ(g) = ∆1(g)m 1 · · ·∆r (g)mr . (4.7)

Set

Ω = g ∈ G : ∆j (g) = 0 for j = 1, . . . , r.

The weight µ is regular if mi = 0 for i = 1, . . . , r. If µ is regular, thenwe see from (4.7) that Ω = g ∈ G : ∆µ(g) = 0. Using techniques orig-inating with Harish-Chandra [H-C], Clerc [Cle] obtained the followingprecise description of the complexified Iwasawa decomposition:

Theorem 4.3 One has Ω = KAN . Let g = k(g)a(g)n(g) be the Iwasawafactorization in G0 .

(i) The function g → n(g) extends holomorphically to a map from Ω toN .

(ii) The functions g → k(g) and g → a(g) extend to multivalent holo-morphic functions on Ω, with values in K and A, respectively.The branches are related by elements of the finite subgroup F =T ∩A.

(iii) Let g → H(g) be the multivalent a-valued function on Ω such thata(g) = expH(g). Then

∆µ(g) = e〈H(g), µ〉 for g ∈ Ω and µ ∈ Λ.

Theorem 4.3 and (4.2) yield a formula analogous to Harish-Chandra’sintegral formula [H-C] for zonal spherical functions on the noncompactsymmetric space G0/K0 :

Corollary 4.2 For g ∈ G let Kg = k ∈ K0 : gk ∈ Ω. Then Kg is anopen set in K0 whose complement has measure zero. For µ ∈ Λ one has

ϕµ(g) =∫

Kg

e〈H(gk), µ〉 dk.

Clerc, elaborating on methods introduced by E. P. Van den Ban[VdBan], uses this integral representation and the method of complexstationary phase to determine the asymptotic behavior of ϕµ(u) as µ →∞ in a suitable cone when u is a regular element of U ; see [Cle, Théorème3.4] for details.


4.3 Horospherical Cauchy–Radon Transform

By Theorem 4.2 the G-modules O[G]R(K ) and O[G]R(M N ) are multiplic-ity free and have the same spectrum (the set Λ of K-spherical highestweights). Using the normalized K-fixed vectors and MN -fixed highestweight vectors, we can thus define bijective G-intertwining maps

T :⊕µ∈Λ

Eµ

∼=→ O[G]R(K ) ,∑µ∈Λ

vµ →∑µ∈Λ

d(µ)〈vµ , πµ∗(g)eKµ∗〉

and

S :⊕µ∈Λ

Eµ

∼=→ O[G]R(M N ) ,∑µ∈Λ

vµ →∑µ∈Λ

〈vµ , πµ∗(g)eµ∗〉.

In both cases we assume that the components vµ = 0 for all but finitelymany µ ∈ Λ.

Let f ∈ O[G]R(K ) . The (algebraic) Peter–Weyl expansion of f is

f(g) =∑µ∈Λ

d(µ)〈vµ , πµ∗(g)eKµ∗〉 (4.8)

where vµ ∈ Eµ and vµ = 0 for all but finitely many µ. Here d(µ) =dim Eµ . Following Gindikin [Gin2], we define the horospherical Cauchy–Radon transform f → f by

f(g) =∑µ∈Λ

〈vµ , πµ∗(g)eµ∗〉

Note that the dimension factor is removed, and the spherical vectoris replaced by the conical vector in Eµ∗ . It is easy to check that thisdefinition does not depend on the choice of spherical and conical vectors,subject to the normalizations (4.4). We can express this transform interms of the maps S and T just introduced as follows: If v ∈

⊕µ Eµ

and f = Tv, then f = Sv. Since S and T are G-module isomorphisms,it follows that the map f → f gives a G-module isomorphism betweenthe function spaces O[G]R(K ) and O[G]R(M N ) . We now express thisisomorphism in a more analytic form.

Theorem 4.4 The horospherical Cauchy–Radon transform is given bythe integral formula

f(g) =∑µ∈Λ

∫U

f(u)∆µ(u−1g) du for g ∈ G (4.9)

(the integrals are zero for all but finitely many µ).

18 Roe Goodman

Remark. The integrands in (4.9) are invariant under u → uk with k ∈K0 , so the integrals can be viewed as taken over the compact symmetricspace X = U/K0 .

Proof Let f be given by (4.8) and let µ ∈ Λ. Since f is right K-invariantand EK

µ = CeKµ , we have∫

U

f(u)πµ(u−1g)eµ du = cµ(g)eKµ (4.10)

for some function cµ(g) on G. From the Schur orthogonality relationsand (4.4) we find that

cµ(g) = 〈vµ , πµ∗(g)eµ∗〉.

Evaluating both sides of (4.10) on the vector eKµ∗ , and summing on µ,

we obtain (4.9).

Remarks. 1. The horospherical Cauchy–Radon transform is the re-presentation-theoretic expression of the double fibration

G

Z = G/K G/MN = Ξ

This sets up a correspondence between Z and Ξ: a point gK of Z mapsto the pseudosphere gKMN ∼= K/M in Ξ, and a point gMN in Ξmaps to the horosphere gMNK ∼= N in Z (see Gindikin [Gin1] for someexamples).

2. Let N = θ(N). Then NMAN is Zariski-dense in G (the generalizedGauss decomposition) and ANK is also Zariski-dense in G (the Iwasawadecomposition). Thus the solvable group AN has an open orbit in G/K

and in G/MN , but the two homogeneous spaces are not isomorphic ascomplex manifolds, even though they have the same G spectrum andmultiplicities.

An invariant (holomorphic) differential operator P (D) on A has apolynomial symbol P (µ) such that

P (D)aµ = P (µ)aµ for µ ∈ Λ.

If µ is a K-spherical highest weight, then the Weyl dimension formula


asserts that

d(µ) =∏α>0

(µ + δ, α)(δ, α)

where δ = 12

∑α>0 α.

Since µ = 0 on t, we can view µ → d(µ) as a polynomial function W (µ)on a∗. Following Gindikin [Gin3], we define the Weyl operator W (D) tobe the differential operator on A with symbol W (µ).

Since A normalizes MN , the space O[G]R(M N ) is stable under R(A).The complex horospherical manifold Ξ is a fiber bundle over the compactflag manifold F = G/MAN (a projective variety), with fiber A. Theoperator W (D) acts by differentiation along the fibers.

Using the Weyl operator, Gindikin [Gin2] obtains the following inver-sion formula for the horospherical Cauchy–Radon transform:

Theorem 4.5 Let f ∈ O[G]R(K ). Then

f(g) =∫

K 0

(W (D)f)(gk) dk for g ∈ G (4.11)

Remark. The integrand in (4.11) is invariant under right translationsby M0 , so the integral is taken over the compact flag manifold K0/M0 =G0/M0A0N0 associated with the dual noncompact symmetric space.

Proof It suffices to prove (4.11) when f(g) = d(µ)〈vµ , πµ∗(g)eKµ∗〉 with

vµ ∈ Eµ . In this case,

f(ga) = 〈vµ , πµ∗(ga)eµ∗〉 = aµ∗〈vµ , πµ∗(g)eµ∗〉.

Hence W (D)f(g) = d(µ)f(g) since d(µ) = d(µ∗). Thus∫K 0

(W (D)f)(gk) dk = d(µ)∫

K 0

〈vµ , πµ∗(g)πµ∗(k)eµ∗〉 dk = f(g),

since the integration of πµ∗(k)eµ∗ over K0 yields eKµ∗ .

4.4 Cauchy–Radon Transform as a Singular Integral

Denote by Z = G/K the complex symmetric space with origin x0 = K.Let ζ0 = MN denote the origin in Ξ. For z = g ·x0 ∈ Z and ζ = y ·ζ0 ∈ Ξwe set ∆j (z | ζ) = ∆j (g−1y). This is well-defined by the transformationproperties (4.5), and we have

∆j (z | ζa) = aµj ∆j (z | ζ) for a ∈ A.

20 Roe Goodman

Following Gindikin ([Gin2], [Gin3]), we define the Cauchy–Radon kernelon Z × Ξ by

K(z | ζ) =∏

1≤j≤l

11−∆j (z | ζ)

.

This function is meromorphic and invariant under the diagonal actionof G, since ∆j (g · z | g · ζ) = ∆j (z | ζ) for g ∈ G. The singular setof K(z | ζ) is the union of the manifolds ∆j (z | ζ) = 1 in Z × Ξ forj = 1, . . . , l.

Recall that X = U/K0 is the compact symmetric space correspondingto θ. Define

Ξ(0) = ζ ∈ Ξ : |∆j (x | ζ)| < 1 for all x ∈ X.

By definition, U · Ξ(0) = Ξ(0). Furthermore, the product formula (4.7)implies that

K(x | ζ) =∑µ∈Λ

∆µ(u−1g) (absolutely convergent series) (4.12)

for x = u·x0 ∈ X and z = g ·ζ0 ∈ Ξ(0). Since A normalizes the subgroupMN , the right multiplication action of A on G gives a right action of A

on Ξ, denoted by ζ, a → ζ ·a. This action commutes with the left actionof G on Ξ.

Lemma 4.3

(i) The map (U/M0)×A → Ξ given by (u, a) → u · ζ0 · a is regular andsurjective.

(ii) Let A+ = a ∈ A : |aµj | < 1 for j = 1, . . . , l. Then U · ζ0 ·A+ ⊂Ξ(0). Hence Ξ(0) is a nonempty open subset of Ξ.

Proof Since U is a maximal compact subgroup of G, the Iwasawa de-composition of G shows that G = UMAN . This implies (i).

Clerc [Cle, Lemme 2.3], using a representation-theoretic argumentoriginating with Harish-Chandra [H-C], shows that |∆µ(u)| ≤ 1 forµ ∈ Λ and u ∈ U . Let a ∈ A. Then Clerc’s estimate implies that

|∆µ(ua)| = |∆µ(u)| |aµ | ≤ |aµ |. (4.13)

If a ∈ A+ then |aµ | < 1. Hence for u, u′ ∈ U we have

|∆µ(u · x0 | u′a · ζ0)| = |∆µ(u−1u′a)| < 1

by (4.13). This implies (ii).


Using Lemma 4.3, we can obtain Gindikin’s singular integral formulafor the horospherical Cauchy-Radon transform. The noncompact realsymmetric space G/U is the space of compact real forms of G, andby the Cartan decomposition of G it is a contractible manifold. Forν = gU ∈ G/U we define a compact totally-real cycle X(ν) = g ·X ⊂ Z

and an open set Ξ(ν)g · Ξ(0) ⊂ Ξ. This furnishes an open covering

Ξ =⋃

ν∈G/U

Ξ(ν)

with a contractible parameter space.

Theorem 4.6 (Gindikin) For f ∈ O[Z] the horospherical Cauchy–Radon transform is given on each set of the covering Ξ(ν) by theCauchy-type singular integral

f(ζ) =∫

X (ν )f(x)K(x | ζ) dx for ζ ∈ Ξ(ν) (4.14)

(the integrand is continuous on X(ν)).

Proof Use formula (4.12) for K(x | ζ) when ζ ∈ Ξ(0), and then translateby g ∈ G to get the formula in general.

5 Concluding Remarks

In this paper we described the harmonic analysis of finitely-transformingfunctions on a compact symmetric space using algebraic group and Liegroup methods, extending the fundamental results of Cartan and Weyl.Our presentation of the horospherical Cauchy-Radon transform has em-phasized groups and homogeneous spaces as in [Gin2]; in fact, the in-tegral formulas hold for all holomorphic functions (not just the G-finitefunctions) on X and Ξ, and also for hyperfunctions. Gindikin’s pointof view is that a compact symmetric space has a canonical dual objectthat is a complex manifold, and he develops this transform emphasizingcomplex analysis and integral geometry (see [Gin3]).

An analytic problem that we have not discussed is the holomorphicextension of real analytic functions on a compact symmetric space.These functions extend holomorphically to complex neighborhoods ofthe space. The geometric and analytic properties of these neighbor-hoods were studied by B. Beers and A. Dragt [Be-Dr], L. Frota-Mattos[Fr-Ma] and M. Lasalle [Las].

22 Roe Goodman

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sentations of Real and p-Adic Groups (Lecture Notes 2, IMS, NationalUniversity of Singapore), World Scientific, 2004.

[Be-Dr] B. L. Beers and A. J. Dragt, New theorems about spherical harmonicexpansions on SU(2), J. Mathematical Phys. 11 (1970), 2313-2328.

[Bor] A. Borel, Essays in the History of Lie Groups and Algebraic Groups (His-tory of Mathematics 21), American Mathematical Society, Providence,2001.

[Bo-Ha] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraicgroups, Annals of Mathematics 75 (1962), 485-535.

[Car1] E. Cartan, Les groupes projectifs qui ne laissent invariante aucunemultiplicité plane, Bull. Soc. Math. de France 41 (1913), 53–96; reprintedin Oeuvres Complètes 1, Part 1, 355–398, Gauthier-Villars, Paris, 1952.

[Car2] E. Cartan, Sur la détermination d’un système orthogonal complet dansun espace de Riemann symétrique clos, Rend. Circ. Mat. Palermo 53(1929), 217-252; reprinted in Oeuvres Complètes 1, Part 2, 1045-1080,Gauthier-Villars, Paris, 1952.

[Cle] J.-L. Clerc, Fonctions sphériques des espaces symétriques compacts,Trans. Amer. Math. Soc. 306 (1988), 421-431.

[Fr-Ma] L. A. Frota-Mattos, The complex-analytic extension of the Fourierseries on Lie groups, in Proceedings of Symposia in Pure Mathematics,Volume 30, Part 2 (1977), 279-282.

[Gin1] S. Gindikin, Holomorphic horospherical duality “sphere-cone”, Indag.Mathem, N.S., 16 (2005), 487-497.

[Gin2] S. Gindikin, Horospherical Cauchy-Radon transform on compact sym-metric spaces, Mosc. Math. J. 6 (2006), no. 2, 299-305, 406.

[Gin3] S. Gindikin, Harmonic analysis on symmetric Stein manifolds from thepoint of view of complex analysis, Jpn. J. Math. 1 (2006), 87-105.

[Go-Wa] R. Goodman and N. R. Wallach, Representations and Invariants ofthe Classical Groups (Encyclopedia of Mathematics and Its Applications,Vol. 68), Cambridge University Press, 1998 (3rd corrected printing 2003).

[H-C] Harish-Chandra, Spherical functions on a semi-simple Lie group. I,Amer. J. Math. 80 (1958), 241-310.

[Haw] T. Hawkins, Emergence of the Theory of Lie Groups: an Essay in theHistory of Mathematics 1869-1926, Springer-Verlag, New York, 2000.

[Hel1] S. Helgason, A duality for symmetric spaces with applications to grouprepresentations, Advances in Math. 5 (1970), 1-154.

[Hel2] S. Helgason, Groups and Geometric Analysis (Pure and Applied Math-ematics 113), Academic Press, Orlando, 1984.

[Hel3] S. Helgason, The Fourier transform on symmetric spaces, in Élie Car-tan et les Mathématiques d’Aujourd’hui, Astérisque No. hors série (1985),Société Mathématique de France, pp. 151-164.

[Las] M. Lasalle, Series de Laurent des fonctions holomorphes dans la com-plexification d’un espace symétrique compact, Ann. Sci. École Norm. Sup.(4) 11 (1978), 167-210.

[Mat] Y. Matsushima, Espaces homogènes de Stein des groupes de Lie com-plexes, Nagoya Math. J. 16 (1960), 205-218.

[Pe-We] F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstel-lungen einer geschlossenen kontinuierlichen Gruppe, Math. Annalen 97(1927), 737-755.


[Ric1] R. W. Richardson, On Orbits of Algebraic Groups and Lie Groups,Bull. Austral. Math. Soc. 25 (1982), 1-28.

[Ric2] R. W. Richardson, Orbits, Invariants, and Representations Associatedto Involutions of Reductive Groups, Invent. math. 66 (1982), 287-313 .

[Ros] M. Rosenlicht, On Quotient Varieties and the Affine Embedding of Cer-tain Homogeneous Spaces, Trans. Amer. Math. Soc. 101 (1961), 211-223.

[VdBan] E. P. Van den Ban, Asymptotic expansions and integral formulas foreigenfunctions on a semisimple Lie group, Thesis, Utrecht, 1983.

[Vi-Ki] E. B. Vinberg and B. N. Kimelfeld, Homogeneous Domains on FlagManifolds and Spherical Subgroups, Func. Anal. Appl. 12 (1978), 168-174.

[Vus] Th. Vust, Opération de groupes réductifs dans un type de cônes presquehomogènes, Bull. Soc. math. France 102 (1974), 317-333.

[Wey1] H. Weyl, Theorie der Darstellung kontinuierlicher halfeinfacher Grup-pen durch lineare Transformationen, I, II, III, und Nachtrag, Math.Zeitschrift 23, 271–309; 24 (1926), 328–376, 377–395, 789–791; reprintedin Selecta Hermann Weyl, 262–366, Birkhäuser Verlag, Basel, 1956.

[Wey2] H. Weyl, Harmonics on homogeneous manifolds, Annals of Mathemat-ics 35 (1934), 486-499; reprinted in Hermann Weyl Gesammelte Abhand-lungen, Band III, 386-399, Springer-Verlag, Berlin - Heidelberg, 1968.

[Wey3] H. Weyl, Elementary algebraic treatment of the quantum mechani-cal symmetry problem, Canadian J. Math. 1 (1949), 57-68; reprinted inHermann Weyl Gesammelte Abhandlungen, Band IV, 346-359, Springer-Verlag, Berlin - Heidelberg, 1968.

[Wey4] H. Weyl, Relativity theory as a stimulus in mathematical research,Proc. Amer. Phil. Soc. 93 (1949), 535-541; reprinted in Hermann WeylGesammelte Abhandlungen, Band IV, 394-400, Springer-Verlag, Berlin -Heidelberg, 1968.

[Žel] D. P. Želobenko, Compact Lie groups and their representations (Trans-lations of mathematical monographs Vol. 40), American MathematicalSociety, Providence, RI, 1973.

2Weyl, eigenfunction expansions and harmonicanalysis on non-compact symmetric spaces

Erik van den BanUniversity of Utrecht

[email protected]

1 Introduction

This text grew out of an attempt to understand a remark by Harish-Chandra in the introduction of [12]. In that paper and its sequel he deter-mined the Plancherel decomposition for Riemannian symmetric spacesof the non-compact type. The associated Plancherel measure turned outto be related to the asymptotic behavior of the so-called zonal spher-ical functions, which are solutions to a system of invariant differentialeigenequations. Harish-Chandra observed: ‘this is reminiscent of a resultof Weyl on ordinary differential equations’, with reference to HermannWeyl’s 1910 paper, [29], on singular Sturm–Liouville operators and theassociated expansions in eigenfunctions.

For Riemannian symmetric spaces of rank one the mentioned systemof equations reduces to a single equation of the singular Sturm–Liouvilletype. Weyl’s result indeed relates asymptotic behavior of eigenfunctionsto the continuous spectral measure but his result is formulated in asetting that does not directly apply.

In [23], Kodaira combined Weyl’s theory with the abstract Hilbertspace theory that had been developed in the 1930’s. This resulted inan efficient derivation of a formula for the spectral measure, previouslyobtained by Titchmarsh. In the same paper Kodaira discussed a class ofexamples that turns out to be general enough to cover all Riemanniansymmetric spaces of rank 1.

It is the purpose of this text to explain the above, and to describelater developments in harmonic analysis on groups and symmetric spaceswhere Weyl’s principle has played an important role.

24

Weyl, eigenfunction expansions, symmetric spaces 25

2 Sturm–Liouville operators

A Sturm–Liouville operator is a second order ordinary differential oper-ator of the form

L = − d

dtp

d

dt+ q, (2.1)

defined on an open interval ] a, b [ , where −∞ ≤ a < b ≤ +∞. Here p isassumed to be a C1-function on ] a, b [ with strictly positive real values;q is assumed to be a real valued continuous function on ] a, b [ .

The operator L is said to be regular at the boundary point a if a isfinite, p extends to a C1-function [ a, b [ → ] 0,∞ [ and q extends to acontinuous function on [a, b [ . Regularity at the second boundary pointb is defined similarly. The operator L is said to be regular if it is regularat both boundary points. In the singular case, no conditions are imposedon the behavior of the functions p and q towards the boundary pointsapart from those already mentioned.

The operator L is formally symmetric in the sense that

〈Lf , g〉[a,b] = 〈f , Lg〉[a,b]

for all compactly supported C2-functions f and g on ] a, b [ . Here wehave denoted the standard L2-inner product on [a, b] by

〈f , g〉[a,b] =∫ b

a

f(t) g(t) dt.

For arbitrary C2-functions f and g on ] a, b [ it follows by partial inte-gration that

〈Lf , g〉[x,y ] − 〈f , Lg〉[x,y ] = [f, g]y − [f, g]x , (2.2)

for all a < x ≤ y < b. Here the sesquilinear form [ · , · ]t on C1( ] a, b [ )(for a < t < b) is defined by

[f, g]t := p(t) [f(t) g′(t)− f ′(t) g(t)]. (2.3)

To better understand the nature of this form, let 〈 · , · 〉 denote the stan-dard Hermitian inner product on C2 , and define the (anti-symmetric)sesquilinear form [ · , · ] on C2 by

[v, w] := 〈Jv , w〉, J =(

0 −11 0

). (2.4)

Define the evaluation map εt : C1( ] a, b [ ) → C2 by

εt(f) := ( f(t) , p(t)f ′(t) ). (2.5)

26 Erik van den Ban

Then the form (2.3) is given by [f, g]t = [ εt(f) , εt(g) ]. We now observethat for ξ a non-zero vector in R2 ,

〈εt(f) , ξ〉 = 0 ⇐⇒ εt(f) ∈ C · Jξ. (2.6)

Hence, if f, g are functions in C1( ] a, b [ ), then by anti-symmetry of theform [ · , · ] we see that

〈εt(f) , ξ〉 = 〈εt(g) , ξ〉 = 0 =⇒ [f, g]t = 0. (2.7)

For a complex number λ ∈ C we denote by Eλ the space of complexvalued C2-functions f on ] a, b [, satisfying the eigenequation Lf = λf.

This eigenequation is equivalent to a system of two linear first orderequations for the function ε(f) : t → εt(f). It follows that for everya < c < b and every v ∈ C2 there is a unique function s(λ, · )v =sc(λ, · )v ∈ C2( ] a, b [ ) such that

s(λ, · )v ∈ Eλ , and εc(s(λ, · )v) = v. (2.8)

By uniqueness, s(λ)v depends linearly on v, and by holomorphic param-eter dependence of the system, the map λ → s(λ)v is entire holomorphicfrom C to C2( ] a, b [ ).

3 The case of a regular operator

After these preliminaries, we recall the theory of eigenfunction expan-sions for a regular Sturm–Liouville operator L on [a, b]. Let ξa , ξb betwo non-zero vectors in R2 . We consider the linear space C2

ξ ([a, b]) ofC2-functions f : [a, b] → C satisfying the homogeneous boundary condi-tions

〈εa(f) , ξa〉 = 0, 〈εb(f) , ξb〉 = 0. (3.1)

For all functions f and g in this space, and for t = a, b, we now have theconclusion of (2.7). In view of (2.2), this implies that L is symmetric onthe domain C2

ξ ([a, b]), i.e.,

〈Lf , g〉[a,b] = 〈f , Lg〉[a,b],

for all f, g ∈ C2ξ ([a, b]). In this setting we have the following result on

eigenfunction expansions. Let σ(L, ξ) be the set of λ ∈ C for which theintersection Eλ,ξ := Eλ ∩ C2

ξ ([a, b]) is non-trivial.


Theorem 3.1 The set σ(L, ξ) is a discrete subset of R without accumu-lation points. For each λ ∈ σ(L, ξ) the space Eλ,ξ is one dimensional.Finally,

L2([a, b]) = ⊕λ∈σ (L,ξ) Eλ,ξ (orthogonal direct sum). (3.2)

We will sketch the proof of this result; this allows us to describe whatwas known about the spectral decomposition associated with a Sturm–Liouville operator when Weyl entered the scene.

For λ ∈ C, let ϕλ be the function in Eλ determined by εa(ϕλ ) = Jξa .

Then 〈εa(ϕλ ) , ξa〉 = 0, hence [ϕλ, ϕλ ]a = 0. The function λ → ϕλ isentire holomorphic with values in C2([a, b]). We observe that Eλ,ξ = 0 ifand only if ϕλ belongs to Eλ,ξ , in which case Eξ ,λ = Cϕλ. We thus see thatthe condition λ ∈ σ(L, ξ) is equivalent to the condition 〈εb(ϕλ ) , ξb〉 = 0.

The function χ : λ → 〈εb(ϕλ ) , ξb〉 is holomorphic with values in C,

and from (2.2) we deduce that

(λ− λ) 〈ϕλ , ϕλ 〉[a,b] = [ϕλ, ϕλ ]b .

In view of (2.7) we now see that the function χ does not vanish forIm λ = 0. Its set of zeros, which equals σ(L, ξ), is therefore a discretesubset of R without accumulation points. Replacing L by a translateL + µ with −µ ∈ R \ σ(L, ξ) if necessary, we see that without loss ofgenerality we may assume that 0 /∈ σ(L, ξ). This implies that L is injec-tive on C2

ξ ([a, b]). Let g ∈ C([a, b]) and consider the equation Lf = g.

Writing this equation as a system of first order equations in terms ofε(f), using a fundamental system for the associated homogeneous equa-tion, and applying variation of the constant one finds a unique solutionf ∈ C2

ξ ([a, b]) to the equation. It is expressed in terms of g by an integraltransform G of the form

Gg(t) =∫ b

a

G(t, τ) g(τ) dτ,

with integral kernel G ∈ C([a, b] × [a, b]), called Green’s function. Theoperator G turns out to be a two-sided inverse to the operator L :C2

ξ ( ] a, b [ ) → C([a, b]).It follows from D. Hilbert’s work on integral equations, [21], that the

map (f, g) → 〈f , Gg〉[a,b] may be viewed as a non-degenerate Hermi-tian form in infinite dimensions, which allows a diagonalization overan orthonormal basis ϕk of L2([a, b]), with associated non-zero diago-nal elements λk , for k ∈ N. In today’s terminology we would say thatthe operator G is symmetric and completely continuous, or compact, and

28 Erik van den Ban

Hilbert’s result has evolved into the spectral theorem for such operators.From this the result follows with σ(L, ξ) = λ−1

k | k ∈ N.

4 The singular Sturm–Liouville operator

We now turn to the more general case of a (possibly) singular operator L

on ] a, b [ . Weyl had written a thesis with Hilbert, leading to the paper[28], generalizing the theory of integral equations to ‘singular kernels.’It was a natural idea to apply this work to singular Sturm–Liouvilleoperators. At the time Weyl started his research it was understoodthat the regular cases involved discrete spectrum. On the other hand,from his work on singular integral equations it had become clear thatcontinuous spectrum had to be expected.

Also, if one considers the example with a = 0, b = ∞, and p = 1, q =0, then L = −d2/dt2 is regular at 0 and singular at ∞. Fix the boundarydatum ξ0 = (0, 1). Then one obtains the eigenfunctions cos

√λt of L,

with eigenvalue λ ≥ 0. In this case a function f ∈ C2c ([0,∞ [ ), satisfying

the boundary condition 〈ε0(f) , ξ0〉 = 0 admits the decomposition

f(t) =∫ ∞

0a(√

λ) cos(√

λt)dλ

π√

λ

involving the continuous spectral measure dλπ√

λ. Here of course, the func-

tion a is given by the cosine transform

a(√

λ) =∫ ∞

0f(t) cos

√λt dt.

Thus, no boundary condition needs to be imposed at infinity. At thetime, Weyl faced the task to unify these phenomena, where both dis-crete and continuous spectrum (in his terminology ‘Punktspektrum’ and‘Streckenspektrum’) could occur, and to clarify the role of the boundaryconditions. Finally, the question arose what could be said of the spectralmeasure.

In [29], Weyl had the important idea to construct a Green operatorfor the eigenvalue problem Lf = λf with λ a non-real eigenvalue. Hefixed boundary conditions for the Green kernel depending on a beautifulgeometric classification of the situation at the boundary points whichwe will now describe. We will essentially follow Weyl’s argument, butin order to postpone choosing bases, we prefer to use the language ofprojective space rather than refer to affine coordinates as Weyl did in[29], p. 226. The reader may consult the appendix for a quick review of


the description of circles in one dimensional complex projective space interms of Hermitian forms of signature type (1, 1).

Returning to the singular Sturm–Liouville problem, we make the fol-lowing observation about real boundary data at a point x ∈ ] a, b [ .

Lemma 4.1 Let λ ∈ C and let f ∈ Eλ\0. Then the following assertionsare equivalent.

(a) ∃ ξ ∈ R2 \ 0 : 〈εx(f) , ξ〉 = 0;(b) [εx(f)] ∈ P1(R);(c) [f, f ]x = 0.

Proof As [f, f ]x = [εx(f), εx(f)], these are basically assertions aboutC2 , which are readily checked.

It follows that the zero set

Cλ,x := f ∈ Eλ | [f, f ]x = 0

defines a circle in the projective space P(Eλ ). Indeed, let εx : P(Eλ ) →P1(C) be the projective isomorphism induced by the evaluation map(2.5), then εx(Cλ,x) = P1(R).

The following important observation is made in Weyl’s paper [29],Satz 1.

Proposition 4.1 Let λ ∈ C \R. Then the circle Cλ,x in P(Eλ ) dependson x ∈ ] a, b [ in a continuous and strictly monotonic fashion. Moreover,if x→ b then Cλ,x tends to either a circle or a point. A similar statementholds for x→ a.

We shall denote by Cλ,b the limit of the set Cλ,x for x → b. Thenotation Cλ,a is introduced in a similar fashion. The proof of the aboveresult is both elegant and simple.

Proof We fix a point c ∈ ] a, b [ . For x ∈ ] c, b [ we define the Hermitianinner product 〈 · , · 〉x on Eλ by 〈f , g〉x = 〈f , g〉[c,x]. It follows from(2.2) that

[f, f ]x = [f, f ]c + 2i Im (λ)〈f , f〉x ,

for f ∈ Eλ and x ∈ ] c, b [ . Without loss of generality, let Im (λ) > 0.

Then it follows that x → −i[f, f ]x is a real valued, strictly increasingcontinuous function. All results follow from this.

30 Erik van den Ban

In his paper [29], Weyl uses a basis f1 , f2 ∈ Eλ such that εc(f2), εc(f1)is the standard basis of C2 . Then [f1 , f2 ]c = 1. In the affine chart deter-mined by f1 , f2 the circle Cλ,c equals the real line. The circles Cλ,x there-fore form a decreasing family of circles which are either all contained inthe upper half plane or in the lower half plane. The form i[ · , · ]x is withrespect to the basis f1 , f2 given by the Hermitian matrix Hkl = i[fk , fl ]x .

It follows that the center of Cλ,x is given by i[f1 , f2 ]x/(−i[f1 , f1 ]x), see(11.2). If Im λ > 0 then the denominator of this expression is positivefor t > c whereas the numerator has limit i for x ↓ c. It follows that inthe affine coordinate z parametrizing f2 + zf1 the circles Cλ,x lie in theupper half plane. Likewise, for Im λ < 0 all circles lie in the lower halfplane.

The limit of Cλ,x as x tends to one of the boundary points is closelyrelated to the L2-behavior of functions from Eλ at that boundary point.

Lemma 4.2 Let λ ∈ C \R, and let f ∈ Eλ \ 0 be such that Cf ∈ Cλ,b .

Then f ∈ L2([c, b [ ) for all c ∈ ] a, b [ .

Proof We may fix a basis f1 , f2 of Eλ such that in the associated affinechart, Cλ,c corresponds to the real line. Then for every x = c the circleCλ,x is entirely contained in the associated affine chart. There exists asequence of points xn ∈ ] c, b [ and Fn ∈ Cλ,xn

such that xn → b andFn → F := Cf.

We agree to write fz = zf1 + f2 . Then there exist unique zn ∈ C suchthat Fn = Cfzn

. Now zn converges to a point z∞ and F = Cfz∞ . Form < n we have

〈fzn, fzn

〉xm≤ 〈fzn

, fzn〉xn

= −(λ− λ)−1 [fzn, fzn

]c .

The expression on the right-hand side has a limit L for n →∞. It followsthat

〈fz∞ , fz∞〉xm≤ L.

This is valid for any m. Taking the limit for xm → b we conclude thatfz∞ ∈ L2([c, b [ ).

Lemma 4.3 Let λ ∈ C \ R and assume that Eλ |[c,b [ ⊂ L2([c, b [ ).

(a) The Hermitian form hλ,x := i[ · , · ]x |Eλhas a limit hλ,b = i[ · , · ]λ,b ,

for x→ b.

(b) The form hλ,b is Hermitian and non-degenerate of signature (1, 1).(c) The limit set Cλ,b is the circle given by [f, f ]λ,b = 0.


(d) In the space Hom(E∗λ , Eλ ) the inverse h−1λ,x converges to h−1

λ,b asx→ b.

Proof (a) From (2.2) it follows by taking the limit for x→ b that

[f, g]λ,b = limx→b

[f, g]x = [f, g]c + (λ− λ)〈f , g〉[c,b]

for all f, g ∈ Eλ . This establishes the existence of the limit hλ,b . As hλ,x

is a Hermitian form for every x ∈ ] a, b [ , the limit is Hermitian as well.(b, d) Fix a basis f1 , f2 of Eλ and write f(z) := z1f1 +z2f2 , for z ∈ C2 .

For x ∈ [c, b] we define the Hermitian matrix Hx by i[f(z), f(w)]x =〈Hxz , w〉. Then Hx → Hb as x→ b. We will finish the proof by showingthat det Hx is a constant function of x ∈ [c, b [ , so that det Hb = det Hc <

0 and moreover H−1x → H−1

b .

Write εx(f) for the linear endomorphism of C2 given by z → εx(f(z)).Then

[f(z), f(w)]x = 〈J εx(f)z , εx(f)w〉 = 〈εx(f)∗J εx(f)z , w〉,

so that Hx = i εx(f)∗Jεx(f). It follows that det Hx = −|det εx(f)|2 . Bya straightforward calculation one sees that det εx(f) = [f1 , f2 ]x . NowLf2 = λf2 , so that from (2.2) it follows that [f1 , f2 ]x − [f1 , f2 ]c = 0 forall x ∈ [c, b [ . Hence det εx(f) = det εc(f) for all x ≥ c.

Finally, the proof of (c) is straightforward.

Combining Lemmas 4.2 and 4.3 we obtain the following corollary.

Corollary 4.4 Let λ ∈ C \R. Then precisely one of the following state-ments is valid.

(a) The limit set Cλ,b is a circle. For any c ∈ ] a, b [ the space Eλ |[c,b [

is contained in L2([c, b [ ).(b) The limit set Cλ,b consists of a single point. For any c ∈ ] a, b [

the intersection of Eλ |[c,b [ with L2([c, b [ ) is one dimensional.

At a later stage in his paper, [29], Satz 5, Weyl used spectral consid-erations to conclude that if (a) holds for a particular non-real eigenvalueλ ∈ C, then Eλ |[c,b [ consists of square integrable functions for any eigen-value λ. We will return to this in the next section, see Lemma 6.2. Itfollows that the validity of (a), and hence the validity of the alternative(b), is independent of the particular choice of the non-real eigenvalue λ.

If (a) holds, the operator L is said to be of limit circle type at b

(‘Grenzkreistypus’), and if (b) holds, L is said to be of limit point type

32 Erik van den Ban

at b (‘Grenzpunkttypus’). With obvious modifications, similar resultsand terminology apply to the other boundary point, a. We note thata regular Sturm–Liouville operator is of the limit circle type at bothboundary points.

Weyl observed that for each boundary point, the type of L determineswhether boundary conditions should be imposed or not. Indeed, if L is ofthe limit point type at the boundary point, then no boundary conditionis needed there. On the other hand, if L is of the limit circle typeat a boundary point, then a boundary condition is required to ensureself-adjointness. Following Weyl, we shall now describe how boundaryconditions can be imposed in the limit circle case.

The idea is to fix a non-real eigenvalue λ ∈ C and to construct a Greenfunction for the operator L− λI. Weyl did this for the particular valueλ = i, but observed that the method works for any choice of non-realλ, see [29], text above Satz 5. In the mentioned paper Weyl considersthe case a = 0, b = ∞, and L regular at a, but the method works ingeneral. In what follows, our treatment will deviate from Weyl’s withregard to technical details. However, in spirit we will stay close to hisoriginal method.

We define D to be the space of functions f ∈ C1( ] a, b [ ) such thatf ′ is locally absolutely continuous (so that Lf is locally integrable).Moreover, we define Db to be the subspace of functions f ∈ D suchthat both f and Lf are square integrable on [c, b [ for some (hence any)c ∈ ] a, b [ . The subspace Da is defined in a similar fashion.

Given two functions f, g ∈ Db , it follows by application of (2.2) that

[f, g]b := limx→b

[f, g]x

exists. If χ ∈ Db , then we denote by Db(χ) the space of functions f ∈ Db

such that [f, χ]b = 0. We now select a non-zero function ϕb,λ ∈ Eλ suchthat the associated point Cϕb,λ ∈ P(Eλ ) belongs to the limit set Cλ,b . Itis possible to characterize the function ϕb,λ by its limit behavior towardsb.

Lemma 4.5

(a) If L is of limit point type at b, then Eλ ∩ Db = Cϕb,λ .

(b) If L is of limit circle type at b, then there exists a function χb ∈ Db

such that Eλ ∩ Db(χb) = Cϕb,λ .

Proof (a) follows from Corollary 4.4. For (b), assume that L is of limit


circle type at b. Then Eλ ⊂ Db . Take χb = ϕb,λ . Then the space on theleft-hand side of the equality equals the space of f ∈ Eλ with [f, χb ]c = 0.

The latter space is one dimensional since [ · , · ]b is non-degenerate onEλ . On the other hand, ϕb,λ belongs to it, by Lemma 4.3, and the resultfollows.

To make the treatment as uniform as possible, we agree to always usethe dummy boundary datum χb = 0 in case L is of limit point type atb. In the limit circle case we select χb as in Lemma 4.5 (b). Then wealways have

Eλ ∩ Db(χb) = Cϕb,λ .

We follow the similar convention for a choice of boundary datum χa ∈Da , so that Da(χa) ∩ Eλ is a line representing a point of the limit setCλ,a . Moreover, we choose a non-zero eigenfunction ϕa,λ spanning thisline. Before proceeding we observe that it follows from Proposition 4.1that

Cλ,a ∩ Cλ,b = ∅.

This implies that ϕa,λ and ϕb,λ form a basis of Eλ . The above choiceshaving been made, we put

Dχ = Da(χa) ∩ Db(χb).

Then Dχ is a subspace of L2( ] a, b [ ). It contains C2c ( ] a, b [ ), hence is

dense. Moreover, it follows from the above that

Dχ ∩ Eλ = 0.

Still under the assumption that λ ∈ C \ R, we now consider the dif-ferential equation

(L− λ)f = g (4.1)

where g is a given square integrable function on ] a, b [ . The equationmay be rewritten as a first order equation for the C2-valued functionε(f). The matrix with columns ε(ϕa,λ ) and ε(ϕb,λ ) is a fundamentalmatrix for this system. By variation of the constant one finds a functionf ∈ D, satisfying (4.1). If g has compact support then f can be uniquelyfixed by imposing the boundary conditions

limx→a

[f, χa ]x = 0, limx→b

[f, χb ]x = 0.

34 Erik van den Ban

This function is expressed in terms of g by means of an integral operator,

f(t) = Gλg(t) :=∫ b

a

Gλ (t, τ) g(τ) dτ, (4.2)

whose integral kernel, called the Green function, is given by

Gλ (t, τ) = w(λ)−1 ϕa,λ(t)ϕb,λ (τ), (t ≤ τ), (4.3)

and Gλ(t, τ) = Gλ (τ, t) for t ≥ τ. Here w(λ) is the Wronskian, definedby

w(λ) = [ϕb,λ , ϕa,λ ]c ,

for a fixed c ∈ ] a, b [ ; note that the expression on the right-hand sideis independent of c, by (2.2). It is an easy matter to show that Gλ iswell defined on L2( ] a, b [ ), with values in D. Moreover, (L−λI)Gλ = I.

Finally, Gλ maps functions with compact support into Dχ .

At this point Weyl essentially proves the following result. He special-izes to λ = i and splits Gλ into real and imaginary part, but the crucialidea is to approximate the Green kernel by Green kernels associated toa regular Sturm–Liouville problem on smaller compact intervals, wherethe spectral decomposition of Theorem 3.1 is applied.

Theorem 4.2 The Green operator Gλ is a bounded linear endomorphismof L2( ] a, b [ ), with operator norm at most |Im λ|−1 .

Proof For z ∈ C we consider the eigenfunction ϕzb = ϕb,λ + zϕa,λ .

As Cϕb,λ is contained in the limit set Cλ,b , there exists a sequence ofpoints bn ∈ ] a, b [ and zn ∈ C such that bn b, zn → 0 and ϕn

b := ϕzn

b

represents a point of the circle Cλ,bn. Similarly, there is a sequence of

points an ∈ ] a, b [ , wn ∈ C such that an a, wn → 0 and ϕna :=

ϕa,λ + wnϕb,λ represents a point of Cλ,an. Define Gn

λ as in (4.3), butwith ϕa,λ and ϕb,λ replaced by ϕn

a and ϕnb respectively. Then it is

readily seen that Gnλ → Gλ, locally uniformly on ] a, b [× ] a, b [ . Apply

the spectral decomposition associated with the regular Sturm–Liouvilleproblem for L on [an , bn ] with boundary data ϕn

a and ϕnb . Then the

operator Gnλ : L2([an , bn ]) → L2([an , bn ]) satisfies (L−λ) Gn

λ = I, hencediagonalizes with eigenvalues (ν − λ)−1 , ν ∈ R. All of these eigenvalueshave length at most |Im λ|−1 , so that ‖Gn

λ ‖ ≤ |Im λ|−1 . It now followsby taking limits that for all f, g ∈ Cc( ] a, b [ ),

|〈f , Gλg〉| = limn→∞

|〈f , Gnλ g〉‖ ≤ | Im λ|.


This implies the result.

Corollary 4.6 The operator Gλ is a bounded linear endomorphism ofthe space L2( ] a, b [ ) with image equal to Dχ . Moreover, Gλ is a two-sidedinverse to the operator L− λI : Dχ → L2( ] a, b [ ).

Proof It is easy to check that Gλ is continuous as a map L2( ] a, b [ ) →C1( ] a, b [ ). Using (2.2) and Theorem 4.2 it is then easy to check thatg → [Gλg, χb ]b is continuous on L2( ] a, b [ ). As this functional vanisheson functions with compact support, it follows that Gλ maps L2( ] a, b [ )into Db(χb). By a similar argument at the other boundary point weconclude that Gλ maps into Dχ .

We observed already that (L − λI)Gλ = I on L2( ] a, b [ ). It followsthat (L−λI) [Gλ (L−λI)−I] = 0 on Dχ . As Gλ maps into Dχ , on whichL− λI is injective, it follows that Gλ(L− λI)− I on Dχ . All assertionsfollow.

Looking at Weyl’s result from a modern perspective, it is now possibleto show that the densely defined operator L with domain Dχ is self-adjoint. To prepare for this, we need a better understanding of theboundary conditions.

In what follows we will assume that L is of limit circle type at b, so thatEλ ⊂ Db . For x ∈ ] a, b [ , the map εx : Eλ → C2 is a linear isomorphism.We define the map βλ,x : Db → Eλ by εx βx(f) = εx(f), for f ∈ Db .

Then βλ,x may be viewed as a projection onto Eλ . For f, g ∈ Db ,

[βλ,x(f), βλ,x(g)]x = [f, g]x . (4.4)

This implies that

[βλ,x(f), · ]x = [f, · ]x on Eλ . (4.5)

As the form [ · , · ]b is non-degenerate on Eλ , we may define a linear mapβλ,b : Db → Eλ by (4.5) with x = b. Then again βλ,b = I on Eλ , so thatβλ,b may be viewed as a projection onto Eλ .

Lemma 4.7 Let f ∈ Db . Then βλ,x(f) → βλ,b(f) in Eλ , as x→ b.

Proof Let γx denote the sesquilinear form [ · , · ]x on Eλ , for a < x ≤ b.

Then for x→ b, we have the limit behavior γx → γb in Hom(Eλ , E∗λ) andγ−1

x → γ−1b in the space Hom(E∗λ , Eλ ), see Lemma 4.3.

36 Erik van den Ban

From (4.5) we deduce that γx(βλ,x(f)) = [f, · ]x , for all f ∈ Db . Itfollows that γx(βλ,x(f)) → [f, · ]b = γb(βλ,b)(f), hence

βλ,x(f) = γ−1x γxβλ,x(f) → βλ,b(f),

for x→ b.

Corollary 4.8 For all f, g ∈ Db we have [f, g]b = [βλ,b(f), βλ,b(g)]b .

Proof This follows from (4.4) by passing to the limit for x→ b.

The following immediate corollary clarifies the nature of the boundarydatum χb.

Corollary 4.9 Let χb ∈ Db . Then Db(χb) depends on χb through itsimage βλ,b(χb) in Eλ .

It follows that in the present setting (L of limit circle type at b), theequality of Lemma 4.5 (b) is equivalent to Cβλ,b(χb) = Cϕb,λ . In otherwords, let Cλ,b denote the preimage in Eλ \ 0 of the limit circle Cλ,b .

Then functions from β−1λ,b(Cλ,b) provide appropriate boundary data at b.

The following result is needed to determine the adjoint of the Greenoperator Gλ . As Eλ ⊂ Db it follows that Eλ = Eλ is contained in Db aswell. We put ϕb,λ := ϕb,λ ; then the above definitions are valid with λ

instead of λ. It is immediate from the definitions that

βλ,b(f) = βλ,b(f), (f ∈ Db). (4.6)

Lemma 4.10

(a) βλ,b βλ,b = βλ,b ;(b) βλ,b(ϕb,λ ) = c ϕb,λ , with c a non-zero complex scalar;(c) βλ,b(χb) = c ϕb,λ ;(d) Db(χb) = Db(χb).

Proof (a) We check that βλ,x = βλ,xβλ,x by applying εx on the left.Now use Lemma 4.7.

(b) Let ψ be an eigenfunction in Eλ . Then [ϕλ,b , ψ]x is constant as afunction of x, by (2.2). Hence, [βλ,b(ϕλ,b), ψ]b = [ϕλ,b , ψ]x . It follows thatβλ,b(ϕλ,b) is non-zero, whereas [βλ,b(ϕλ,b), ϕλ,b ]b = [ϕλ,b , ϕλ,x ]x = 0. Theassertion follows.

(c) Applying (4.6), (a) and (b) we obtain: βλ,b(χb) = βλ,b(χb) =


βλ,bβλ,b(χb) = βλ (ϕb,λ ) = cϕb,λ . Finally, (d) is an immediate conse-quence of (c).

We now return to the situation of a general singular Sturm–Liouvilleoperator L.

Theorem 4.3 The operator L with domain Dχ is self-adjoint.

Proof From Lemma 4.10 it follows that Dχ = Dχ . The adjoint G∗ of theoperator G = Gλ has integral kernel G∗

λ (t, τ) := Gλ (τ, t). This is preciselythe Green kernel associated with the eigenvalue λ and the boundary dataχa , χb . As its image Dχ equals Dχ , it follows that G∗ is the two-sidedinverse of the bijection L − λI : Dχ → L2( ] a, b [ ). These facts implythat the adjoint L∗ equals L.

At this point one can prove the following generalization of Theorem3.1, due to Weyl, [29], Satz 4. Let σ(L,χ) be the set of λ ∈ C for whichEλ ∩ Dχ = 0.

Theorem 4.4 (Weyl 1910) Let L be of limit circle type at both endpoints. Then σ(L,χ) is a discrete subset of R, without accumulationpoints. Moreover, L2( ] a, b [ ) is the orthogonal direct sum of the spacesEλ ∩ Dχ .

Proof Weyl proved this by using the Green operator G correspondingto the eigenvalue i. Let G2 be the imaginary part of its kernel. Then G2

is real valued, symmetric and square integrable, hence admits a diago-nalization. In today’s terminology, the associated integral operator G2 ,

which equals (2i)−1 [G − G∗], is self-adjoint and Hilbert-Schmidt, hencecompact. All its eigenspaces are finite dimensional, and contained in Dχ ,

since Dχ = Dχ . Moreover, each of them is invariant under the symmetricoperator L.

The regular case may be viewed as a special case of the above. Indeed,if ξa , ξb are the boundary data of Theorem 3.1, let µ ∈ C\R be arbitrary,and for x = a, b, let χx be the constant function with value Jξx. ThenEλ,ξ = Eλ ∩ Dχ , for all λ ∈ C.

5 Weyl’s spectral theorem

Using Green’s function Gλ for non-real λ and his earlier work on singularintegral equations, [28], Weyl was able to establish the existence of a

38 Erik van den Ban

spectral decomposition of L2( ] a, b [ ) in terms of eigenfunctions of L withreal eigenvalue. In [29] he considers the case a = 0, b = ∞, and assumesthat L is regular (hence of circle limit type) at 0. Let ξa ∈ R2 \ 0be a boundary datum at a, and fix a unit vector η ∈ R2 perpendicularto ξ. Let χ0 be the constant function with value η and let χ∞ be aboundary datum at ∞ (if L is of limit point type at ∞, we take thedummy boundary datum χ∞ = 0). For each λ ∈ C let ϕλ be the uniqueeigenfunction of L with eigenvalue λ and ϕλ(a) = η. Then accordingto Weyl, [29], Satz 5,7, there exists a right-continuous monotonicallyincreasing function ρ such that each function f ∈ C2( ] 0,∞ [ ) ∩ Dχ

admits a decomposition of the form

f(x) =∫

R

ϕλ(x) dF (λ) (5.1)

with uniformly and absolutely converging integral; here dF (λ) is a reg-ular Borel measure, defined by

dF (∆) =∫ ∞

0f(t)

∫∆

ϕλ(t) dρ(λ) dt. (5.2)

In the above, dρ denotes the regular Borel measure determined by theformula dρ( ]µ, ν]) = ρ(ν)− ρ(µ), for all µ < ν.

Actually, Weyl’s original formulation was different and involved a dis-crete and a continuous part. His formulation follows from the one aboveby the observation that ρ admits a unique decomposition ρ = ρd + ρc

with ρc a continuous monotonically increasing function with ρc(0) = 0,and with ρd a right-continuous monotonically increasing function whichis constant on each interval where it is continuous.

In case L is of the limit circle type at infinity, the decomposition isdiscrete by Theorem 4.4, so that ρc = 0, so that the above gives rise toa discrete decomposition. In case L is of the limit point type at ∞, thedecomposition is of mixed discrete and continuous type.

It has now become customary to write

dF (λ) = Ff(λ) dρ(λ), Ff(λ) =∫ ∞

0f(t)ϕλ (t) dt, (5.3)

with the interpretation that the integral converges as an integral withvalues in L2(R, dρ).

We will call ρ the spectral function associated with the operator L,

the boundary data χ0 , χ∞, and the choice of eigenfunctions ϕλ. In [29],Weyl also addressed the natural problem to determine its continuouspart ρc .


Theorem 5.1 (Weyl 1910) Assume L is a Sturm–Liouville operator ofthe form (2.1) on [0,∞ [ , regular at 0. Assume moreover that thecoefficients p and q satisfy the conditions

(a) limt→∞ t|p(t)− 1| = 0, limt→∞ t q(t) = 0,(b)

∫∞0 t|p(t)− 1| dt < ∞,

∫∞0 t|q(t)| < ∞.

Then L is of the limit point type at ∞. Let ξa , η, χa and ϕλ be defined asabove and let ρ be the associated spectral function. Then the support ofdρd is finite and contained in the open negative real half line ] −∞, 0 [ .The support of dρc is contained in the closed positive real half line [0,∞ [.There exist uniquely determined continuous functions a, b : ] 0,∞ [→ Rsuch that

ϕλ(t) = a(λ) cos(t√

λ) + b(λ) sin(t√

λ) + o(t). (5.4)

In terms of these coefficients, the spectral measure dρc is given by

dρc(λ) =1

a(λ)2 + b(λ)2

dλ

π√

λ.

Here we note that by (5.4) and the condition on f, the integral (5.3)is absolutely convergent. If p = 1 and q = 0, then of course one hasa(λ) = η1 and b(λ) = η2 , and one retrieves the continuous measuredρc(λ) = (π

√λ)−1dλ.

Let c(√

λ) := 12 (a(λ)− ib(λ)). Then

ϕλ (t) = c(λ)eit√

λ + c(λ)e−it√

λ + o(t)

and the spectral measure is given by

dρ(λ) =d√

λ

2π|c(√

λ)|2(5.5)

We may view the operator L as a perturbation of the operator −d2/dt2 .

At infinity the eigenfunction ϕλ behaves asymptotically as a linear com-bination of the exponential eigenfunctions for the unperturbed problem,with amplitudes of equal modulus |c(

√λ)|. The spectral measure of the

perturbed problem is obtained from the spectral measure of the un-perturbed problem by dividing through |c(

√λ)|2 . As we will see later,

this principle is omnipresent in the theory of harmonic analysis of non-compact Riemannian symmetric spaces, of non-compact real semisimpleLie groups, and of their common generalization, the so-called semisimplesymmetric spaces.

40 Erik van den Ban

6 Dependence on the eigenvalue parameter

In this section we will prove holomorphic dependence of the Green func-tion Gλ on the parameter λ. This is not obvious from the definition(4.3). Indeed, in the limit circle case at b, the particular normalizationof ϕb,λ chosen only guarantees real analytic dependence on the param-eter λ (this fairly easy result will not be needed in the sequel). In thelimit point case, only the line Cϕb,λ does not depend on the choicesmade, but the dependence of ϕb,λ on λ may be arbitrary. The followingresult suggests to look for differently normalized eigenfunctions, whichdo depend holomorphically on λ.

Lemma 6.1 The Green kernel Gλ defined by (4.2) depends on ϕa,λ andϕb,λ through their images in P(Eλ ).

Proof This is caused by the division by the Wronskian w(λ)=[ϕb,λ , ϕa,λ ].

The following result follows by application of the method of variationof the constant as explained in [7], Thm. 2.1, p. 225. The assertionabout holomorphic dependence is not given there, but follows by thesame method of proof.

Lemma 6.2 Let a < c < b and assume that for some λ0 ∈ C theeigenspace Eλ0 |[c,b [ is contained in L2([c, b [ ). Then for each eigenvalueλ ∈ C the associated eigenspace Eλ |[c,b [ is contained in L2([c, b [ ).

Moreover, for each c ∈ ] a, b [ and all v ∈ C2 , the function λ →sc(λ, · )v|[c,b [ (see (2.8)) is entire holomorphic as a function with valuesin L2([c, b [ ).

In the following, we assume that L is of limit circle type at b. Fromthe text below (4.5) we recall the definition of the map βλ,b : Db → Eλ ,

for every λ ∈ C \ R.

Lemma 6.3 Let L be of the limit circle type at b. Then for all µ, λ ∈C \ R,

(a) βµ,b βλ,b = βµ,b ;(b) the restriction βµ,b |Eλ

is a linear isomorphism onto Eµ ;(c) the restriction βµ,b |Eλ

induces a projective isomorphism P(Eλ ) →P(Eµ), mapping the limit circle Cλ,b onto the limit circle Cµ,b .


Proof Assertion (a) is proved in the same fashion as assertion (a) ofLemma 4.10. Since [ · , · ]b is non-degenerate on both Eλ and Eµ , assertion(b) follows by application of Corollary 4.8. Finally, (c) follows from theidentity of Corollary 4.8, in view of Lemma 4.3.

The following result suggests the modification of the eigenfunctions in(4.2) that we are looking for.

Lemma 6.4 Let L be of the limit circle type at b. Let χb ∈ Db andassume that for some µ ∈ C \ R the function βµ,b(χb) is non-zero andrepresents a point of the limit circle Cµ,b . Then

(a) for each λ ∈ C\R the function βλ,b(χ) is a non-zero eigenfunctionin Eλ which represents a point of the limit circle Cλ,b ;

(b) for each c ∈ ] a, b [ , the map λ → [εc(βλ,b(f))] is holomorphicfrom C \ R to P1(C) \ P1(R).

Proof By the first assertion of Lemma 6.3, βλ,b(χ) = βλ,bβµ,b(χ) =βλ,b(ϕb,µ). Assertion (a) follows by application of the remaining asser-tions of the mentioned lemma.

We now turn to (b). We will prove the holomorphy in a neighborhoodof the fixed point λ0 ∈ C \ R. As βλ,b(χ) = βλ,b(βλ0 ,b(χ)) we may aswell assume that χ ∈ Eλ0 and that [χ] ∈ Cλ0 ,b . Select a sequence xn

in ] a, b [ converging to b, and for each n a point pn ∈ Cλ0 ,xnsuch that

pn → [χ]. There exist χn ∈ Eλ0 such that [χn ] = pn and χn → χ inEλ0 . We define ϕλ,n ∈ Eλ by εxn

ϕn,λ = εxnχn . Then in the notation of

(4.5), ϕn (λ) equals βλ,xn(χn ) and represents a point of Cλ,xn

. For eachfixed λ the sequence βλ,xn

|Eλ 0in Hom(Eλ0 , Eλ ) has limit βλ,b . Hence

ϕn (λ) → βλ,b(χ), pointwise in λ.

Let c ∈ ] a, b [ . Passing to a subsequence we may assume that xn > c

for all n ≥ 1. The map εc : P(Eλ ) → P1(C) maps the circle Cλ,c ontoP1(R). Let Ω be a connected open neighborhood of λ0 . Then it followsby application of Proposition 4.1 that all circles εc(Cλ,xn

) are containedin one particular connected component U of P1(C)\P1(R). This impliesthat ψn (λ) := εcβxn ,λ(f) ∈ U for every λ ∈ Ω. By using an affinechart containing the compact closure of U we see that the sequence ψn

has a subsequence converging locally uniformly to a holomorphic limitfunction ψ : Ω → U. By pointwise convergence, ψ(λ) = [εcβλ,b(χ)], and(b) follows.

42 Erik van den Ban

Corollary 6.5 Let L be of the limit circle type at b and let χb ∈ Db be asin the above lemma. There exists a family of functions ϕλ,b ∈ C2( ] a, b [ )depending holomorphically on λ ∈ C \ R such that for each λ ∈ C \ R

(a) ϕb,λ ∈ Eλ \ 0;(b) ϕb,λ represents the point [βλ,b(χb)] of the limit circle Cλ,b .

The following analogous result in the limit point case can be provedusing a similar method, see [7], Thm. 2.3, p. 229, for details.

Lemma 6.6 Let L be of the limit point type at b. Then there exists afamily of functions ϕb,λ ∈ C2( ] a, b [ ), depending holomorphically on theparameter λ ∈ C \ R, such that for each λ ∈ C \ R,

(a) ϕb,λ ∈ Eλ \ 0;(b) the function ϕb,λ represents the limit point in P(Eλ ).

Let L be arbitrary again. We fix boundary data χa, χa as indicatedin the previous section, so that L : Dχ → L2( ] a, b [ ) is self-adjoint.Accordingly, we fix holomorphic families of eigenfunctions ϕa,λ , ϕb,λ ∈Eλ in the manner indicated in Corollary 6.5 and Lemma 6.6.

Finally, we define the Green function Gλ by means of the formula(4.3). The functions ϕa,λ , ϕb,λ used here are renormalizations of thoseused in Section 4. By Lemma 6.1 this does not affect the definition ofthe Green kernel.

Corollary 6.7 The Green kernel Gλ ∈ C( ] a, b [× ] a, b [ ) depends holo-morphically on the parameter λ ∈ C \ R.

This result of course realizes the resolvent (L−λI)−1 of the self-adjointoperator L with domain Dχ explicitly as an integral operator with kerneldepending holomorphically on λ.

7 A paper of Kodaira

For the general singular Sturm–Liouville problem, there exists a spec-tral decomposition similar to (5.1), but with a spectral matrix insteadof the spectral function ρ. Weyl observed this in [30]. The spectral ma-trix was later determined by E.C. Titchmarsh who used involved directcomputations using the calculus of residues, see [27].

Independently, K. Kodaira [23] rediscovered the result by a very el-egant method, combining Weyl’s construction of the Green functionwith the general spectral theory for self-adjoint unbounded operators


on Hilbert space, as developed in the 1930’s by J. von Neumann and M.Stone. Weyl was very content with this work of Kodaira, as becomesclear from the following quote from the Gibbs lecture delivered in 1948,[31], p. 124: ‘The formula (7.5) was rediscovered by Kunihiko Kodaira(who of course had been cut off from our Western mathematical liter-ature since the end of 1941); his construction of ρ and his proofs for(7.5) and the expansion formula [...], still unpublished, seem to clinchthe issue. It is remarkable that forty years had to pass before such athoroughly satisfactory direct treatment emerged; the fact is a reflectionon the degree to which mathematicians during this period got absorbedin abstract generalizations and lost sight of their task of finishing upsome of the more concrete problems of undeniable importance.’

We will now describe the spectral decomposition essentially as pre-sented by Kodaira [23]. Fix boundary data χa and χb as in Theorem4.3. We use the notation H := L2( ] a, b [ ). Then the operator L withdomain Dχ is a self-adjoint operator in the Hilbert space H; it thereforehas a spectral resolution dE.

To obtain a suitable parametrization of the space of eigenfunctionsfor L, fix c ∈ ] a, b [ and recall that the map εc : f → (f(c), p(c)f ′(c))is a linear isomorphism from Eλ onto C2 , for each λ ∈ C. We definethe function s(λ) = sc(λ, · ) : ] a, b [→ Hom(C2 , C) as in (2.6). Thenλ → s(λ) may be viewed as an entire holomorphic map with values inC2( ] a, b [ )⊗Hom(C2 , C)). Moreover, for each λ ∈ C the map v → s(λ)vis a linear isomorphism from C2 onto Eλ .

For f ∈ Cc( ] a, b [ ) and λ ∈ R we define the Fourier transform

Ff(λ) =∫ b

a

s(λ, x)∗f(x) dx, (7.1)

where s(λ, x)∗ ∈ Hom(C, C2) is the adjoint of s(λ, x) with respect to thestandard Hermitian inner products on C2 and C.

By a spectral matrix we shall mean a function P : R → End(C2) withthe following properties

(a) P (x)∗ = P (x), i.e., P (x) is Hermitian with respect to the standardinner product, for all x ∈ R;

(b) P is continuous from the right;(c) P (0) = 0 and P (y)− P (x) is positive semi-definite for all x ≤ y.

Associated with a spectral matrix as above there is a unique regularBorel measure dP on R, with values in the space of positive semi-definiteHermitian endomorphisms of C2 , such that dP ( ]µ, ν]) = P (ν) − P (µ)

44 Erik van den Ban

for all µ ≤ ν. Conversely, a measure with these properties comes from aunique spectral matrix P. Given a spectral matrix P, we define M2 =M2,P to be the space of Borel measurable functions ϕ : R → C2 with

〈ϕ , ϕ〉P :=∫

R

〈ϕ(ν) , dP (ν)ϕ(ν)〉 < ∞.

Moreover, we define H = HP to be the Hilbert space completion of thequotient M2/M⊥

2 .

Let Tλ := εc Gλ . Then Tλ : H → C2 is a continuous linear map. Wedenote its adjoint by T ∗

λ . Kodaira uses the elements γ1(λ), γ2(λ) of Hdetermined by prj Tλ = 〈 · , γj (λ)〉.

Theorem 7.1 (Kodaira 1949) The spectral function P determined by

dP (ν) = |ν − λ|2 Tλ dE(ν) T ∗λ (7.2)

is independent of λ ∈ C \ R. Moreover, it has the following properties.

(a) The Fourier transform extends to an isometry from the Hilbertspace H = L2( ] a, b [ ) onto the Hilbert space H = HP .

(b) The spectral resolution dE(ν) of the self-adjoint operator L withdomain Dχ is given by

F dE(S) = 1S F ,

for every Borel measurable set S ⊂ R; here 1S denotes the mapinduced by multiplication with the characteristic function of S.

For the proof of Theorem 7.1, which involves ideas of Weyl [29], we re-fer the reader to Kodaira’s paper [23]. In addition to the above, Kodairaproves more precise statements about the nature of the convergence ofthe integrals in the associated inversion formula.

After having introduced the spectral matrix, Kodaira gives an inge-nious short proof of an expression for the spectral matrix which had beenfound earlier by Titchmarsh. We observe that the C2-valued functionsFa(λ) = εc(ϕa,λ) and Fb(λ) = εc(ϕb,λ ) are holomorphic functions ofλ ∈ C \R. The matrix F (λ) with columns Fa(λ) and Fb(λ) is invertiblefor λ ∈ C \ R. By the above definitions,

ϕa,λ(t) = s(λ, t)Fa(λ), ϕb,λ (t) = s(λ, t)Fb(λ). (7.3)

We now define the 2×2 matrix M(λ), the so-called characteristic matrix,


by M(λ) = −(det F )−1FaFTb , i.e.,

M(λ) = −det F (λ)−1(

Fa1Fb1 Fa1Fb2

Fa2Fb1 Fa2Fb2

)λ

(7.4)

Actually, Kodaira uses the symmetric matrix M(λ)− 12 J, which has the

same imaginary part. The matrix M(λ) depends holomorphically on theparameter λ ∈ C \ R.

Theorem 7.2 (Titchmarsh, Kodaira) The spectral matrix P is given bythe following limit:

P (ν) = limδ↓0

limε↓0

1π

∫[δ,ν+δ ]+iε

Im M(λ) dλ. (7.5)

Proof Multiplying both sides of (7.2) with |ν−λ|−2 and integrating overR, we find that ∫

R

|ν − λ|−2 dP (ν) = TλT ∗λ .

By a straightforward, but somewhat tedious calculation, using (2.2) and[ϕa,λ , ϕa,λ ]a = [ϕb,λ , ϕb,λ ]b = 0, it follows that

Im λ TλT ∗λ =

12i

1|[Fb, F a ]|2

( [Fa, Fa ]FbFTb − [Fb, Fb ]FaF

Ta ).

This in turn implies that Im λ · TλT ∗λ = ImM(λ). Hence,∫

R

|ν − λ|−2Im λ dP (ν) = Im M(λ).

From this (7.5) follows by a straightforward argument.

After this, Kodaira shows that the above result can be extended to amore general basis of eigenfunctions. A fundamental system for L is alinear map s(λ) : C2 → Eλ , depending entire holomorphically on λ ∈ Cas a C2( ] a, b [ )-valued function, such that the following conditions arefulfilled for all λ ∈ C :

(a) s(λ)v = s(λ)(v), (v ∈ C2);(b) det(εx s(λ)) = 1, (x ∈ ] a, b [ ).

Put sj (λ) = s(λ)ej , then condition (b) means precisely that the Wron-skian [s1(λ), s2(λ)]x equals 1. Write ψ(λ) = εc s(λ) ∈ End(C2). Thenit follows that ψ entire holomorphic, and that det ψ(λ) = 1. Moreover,

s(λ, x) = sc(λ, x)ψ(λ).

46 Erik van den Ban

We may define the Fourier transform associated with s by the identity(7.1). The associated spectral function P is expressed in terms of thespectral matrix Pc for sc by the equation

dP (λ) = ψ(λ)−1 dPc(λ) ψ(λ)∗−1 .

We define the matrix F for s by the identity (7.3). Then the associatedmatrix M, defined by (7.4) is given by

M(λ) = ψ(λ)−1 Mc(λ) ψ(λ)T−1 .

Kodaira shows that with these definitions, the identity (7.5) is still valid.

8 A special equation

In the second half of the paper [23], Kodaira applies the above resultsto the time independent one dimensional Schrödinger operator

L = − d2

dt2+ m(m + 1)t−2 + V (t),

with m ≥ − 12 and tV (t) a real valued real analytic function on an open

neighborhood of [0,∞ [ , such that

tV (t) = O(t−ε), for t →∞,

with ε > 0. Actually, Kodaira considers a more general problem withweaker requirements both at infinity and zero, but we shall not needthis. It is in fact not clear that his condition on the behavior of V

at 0 is strong enough for the subsequent argument to be valid, as waspointed out by [22], p. 206. Kodaira’s argumentation, which we shall nowpresent, is valid under the hypotheses stated above, as they imply thatthe eigenequation Lf = λf has a regular singularity at zero. Becauseof this, the asymptotic behavior of the eigenfunctions towards zero iscompletely understood. Indeed, the associated indicial equation hassolutions m + 1 and −m, where m + 1 ≥ −m. Let c0 be any non-zeroreal constant. Then there exists a unique eigenfunction s1(λ) ∈ Eλ suchthat

s1(λ, t) = c0 tm+1ϕ(λ, t)

with ϕ(λ, · ) real analytic in an open neighborhood of 0 and ϕ(λ, 0) = 1.It can be shown that ϕ(λ, t) is entire holomorphic in λ and real valued forreal λ. Kodaira claims that there exists a second eigenfunction s2(λ) ∈Eλ , depending holomorphically on λ, such that s1 , s2 form a fundamental


system fulfilling the requirements (a) and (b) stated below Theorem 7.2.Using the theory of second order differential equations with a regularsingularity this can indeed be proved along the following lines.

If k := (m + 1) − (−m) = 2m + 1 is strictly positive, there exists asecond eigenfunction s2(λ) ∈ Eλ with

s2(λ, t) = −c−10 (2m + 1)−1 t−m +O(t−m+ε), (t → 0).

If k is not an integer, this eigenfunction is unique. If k is an integer, thens2(λ) has a series expansion in terms of t−m+r and tm+1+s log t, (r, s ∈N), and is uniquely determined by the requirement that the coefficientof t−m+k = tm+1 is zero. Finally, if k = 0, i.e., m = − 1

2 , then thereexists a unique second eigenfunction s2(λ, t) with

s2(λ, t) = c−10 t1/2 log t +O(t1/2+ε), (t → 0).

In all cases, by arguments involving monodromy for t around zero itcan be shown that s2(λ, t) is entire holomorphic in λ and real valuedfor real λ. Finally, from the series expansions for these functions andtheir derivatives, it follows that the Wronskian [s1(λ), s2(λ)]t behaveslike 1 + O(tε) for t → 0. Since the Wronskian is constant, this impliesthat s1 , s2 is a fundamental system.

From the asymptotic behavior of s1 , s2 it is seen that at the boundarypoint 0, the operator L is of limit circle type if and only if m > 1

2 . It isof limit point type if − 1

2 ≤ m ≤ 12 . In the first case we fix the boundary

datum χ0 = s1(0, · ) at 0 and in the second case we fix the (dummy)boundary datum χ0 = 0. In all cases s1 is square integrable on ] 0, 1], sothat ϕ0λ = s1(λ) and F0(λ) = (1, 0)T , in the notation of (7.3).

We now turn to the asymptotic behavior at ∞. Kodaira first showsthat for every ν with Im ν ≥ 0, ν = 0, there is a unique solution Φν tothe equation Lf = ν2f such that

Φν (t) ∼ eiν t , (t →∞),

the asymptotics being preserved if the expressions on both sides aredifferentiated once with respect to t. Moreover, both Φν (t) and Φ′

ν (t)are continuous in (t, ν) and holomorphic in ν for Im ν > 0.

For Im ν < 0 the function Ψν = Φν belongs to Eν 2 and Ψν (t) ∼ e−iν t

for t →∞. This shows that Ψν is not square integrable towards infinity,so that L is of limit point type at infinity. We may therefore take

ϕ∞λ = Φν , (Im λ > 0, Im ν > 0, ν2 = λ).

It follows from the above that Φν (t) = a(ν) s1(ν2 , t)+b(ν) s2(ν2 , t), with

48 Erik van den Ban

a, b continuous on Im ν ≥ 0, ν = 0, and holomorphic on Im ν > 0. Wenote that F∞(λ) = (a(ν), b(ν))T . Using the similar expression for Φ−ν

it follows that

a(ν) = a(−ν), b(ν) = b(−ν). (8.1)

If ν is real and non-zero, then Φν and Φ−ν form a basis of Eν 2 andfrom the asymptotic behavior of the (constant) Wronskian [Φν ,Φ−ν ]tone reads off that

b(ν) a(−ν)− a(ν) b(−ν) = 2iν, (ν ∈ R \ 0). (8.2)

From (8.1) and (8.2) it follows that

Im a(ν)b(ν) = −ν, (ν ∈ R \ 0). (8.3)

In particular, a and b do not vanish anywhere on R \ 0.We can now determine the spectral matrix for this problem. Indeed,

for Im λ > 0 and Im ν > 0, ν2 = λ,

F (λ) =(

1 a(ν)0 b(ν)

),

so that

Im M(λ) = −Im(

a(ν)b(ν)−1 10 0

)=

(−Im a(ν )

b(ν ) 00 0

)by (7.4). From this we conclude that the spectral matrix P (λ) has zeroentries except for the one in the upper left corner, which we denote byρ(λ). The second component of Ff now plays no role in the Plancherelformula. Indeed, define

F1f(λ) =∫ ∞

0f(t) s1(λ, t) dt,

then we have the following.

Corollary 8.1 F1 extends to an isometry from the space L2( ] 0,∞ [ )onto L2(R, dρ). The spectral function ρ is given by

ρ(λ) = − 1π

limδ↓0

limε↓0

∫[δ,λ+δ ]+iε

Ima(√

µ)b(√

µ)dµ, (8.4)

where the square root√

µ with positive imaginary part should be taken.

Since a and b are holomorphic in the upper half plane, a(√

µ) b(√

µ)−1

is meromorphic over the interval ] −∞, 0 [ , so that on ] −∞, 0 [ , the


measure dρ is a countable sum of point measures. Indeed, let S be the(discrete) subset of zeros for a on the positive imaginary axis i ] 0,∞ [ .Then

dρ| ] −∞,0 [ =∑σ∈S

2Resν=σν a(ν)b(ν)

· δ[σ 2 ]

On the other hand, for λ > 0, if µ → λ, then the integrand of (8.4) tendsto Im a(

√λ)b(

√λ)−1 , with local uniformity in λ. In view of (8.3) it now

follows that

dρ(λ)| ] 0,∞ [ = − 1π

Ima(√

λ)b(√

λ)=

1π

√λ dλ

|b(√

λ)|2=

2π

λ d√

λ

|b(√

λ)|2.

Finally, if s1(0) is not square integrable at infinity, then ρ0 := dρ(0) =0. On the other hand, if it is, then ρ0 := dρ(0) equals the squaredL2-norm of s1(0).

Finally, since s1(0, λ) is real valued for λ real, whereas Φ−ν = Φν forreal ν, there exists a real analytic function c : R \ 0 → C such that

s1(0, ν2) = c(ν)Φν + c(−ν)Φ−ν

for all ν ∈ R \ 0. This gives rise to the equationsa(ν)c(ν) + a(−ν)c(−ν) = 1b(ν)c(ν) + b(−ν)c(−ν) = 0.

Using (8.2) we now deduce that

c(ν) = −b(ν)/2iν, (ν ∈ R \ 0). (8.5)

Therefore,

dρ(λ)| ] 0,∞ [ =12π

d√

λ

|c(√

λ)|2.

We thus see that the principle formulated below (5.5) still holds in thissetting.

9 Riemannian symmetric spaces

A Riemannian symmetric space is a connected Riemannian manifold X

with the property that the local geodesic reflection at each point extendsto a global isometry of X. Up to covering, each such space allows adecomposition into a product of three types of symmetric space. Thosewith zero sectional curvature (the Euclidean spaces), those with positive

50 Erik van den Ban

sectional curvature (among which the Euclidean spheres) and those withnegative sectional curvature (among which the hyperbolic spaces). Itfollows from the work of E. Cartan, that the spaces of negative sectionalcurvature are precisely those given by X = G/K, where G is a connectedreal semisimple Lie group of non-compact type, with finite center, andwhere K is a maximal compact subgroup of G. The Killing form of G

naturally induces a G-invariant Riemannian metric on G/K. The groupK is the fixed point group of a Cartan involution θ of G; this involutioninduces the geodesic reflection in the origin e = eK of X.

A typical example of a symmetric space of this type is the space X ofpositive definite symmetric n× n-matrices on which G = SL(n, R) actsby (g, h) → ghgT . The stabilizer of the identity matrix equals SO(n)and the associated Cartan involution θ : G → G is given by g → (gT)−1 .

The geodesic reflection in the identity matrix I is given by h → h−1 .

In the general setting, the derivative of the Cartan involution at theidentity element of G induces an involution θ∗ of the Lie algebra g. TheLie algebra g decomposes as a direct sum of vector spaces

g = k⊕ p,

where k and p are the +1 and −1 eigenspaces of θ∗, respectively. It canbe shown that the map

(X, k) → expXk, p×K → G (9.1)

is an analytic diffeomorphism onto G. In particular, this implies thatthe exponential map induces a diffeomorphism Exp : X → exp XK,

p → G/K. Let a be a subspace of p, maximal subject to the conditionthat it is abelian for the Lie bracket of g. Every other such subspace is K-conjugate to a. The dimension r of a is called the rank of the symmetricspace G/K.

In the example G = SL(n, R), the Lie algebra sl(n, R) consists ofall traceless n× n-matrices, and θ∗ is given by X → −XT . The Cartandecomposition (9.1) is given by the decomposition of a matrix in terms ofa positive definite symmetric one times an orthogonal one. The algebraa now consists of the traceless diagonal matrices, so that the rank ofSL(n, R)/SO(n) equals n − 1. For n = 2 the space is isomorphic tothe hyperbolic upper half plane, equipped with the action of SL(2, R)through fractional linear transformations.

By a result of Harish-Chandra, the algebra D(G/K) of G-invariantlinear partial differential operators on G/K is a polynomial algebraof rank r. More precisely, let M be the centralizer of a in K, and let


W := NK (a)/M, the normalizer modulo the centralizer of a in K. As asubgroup of GL(a), this group is the reflection group associated with theroots of a in g. It is therefore called the Weyl group of the pair (g, a).There exists a canonical isomorphism γ from D(G/K) onto P (a∗

C)W , the

algebra of W -invariants in the polynomial algebra of the complexifieddual space a∗

C(equipped with the dualized Weyl group action). By a

result of C. Chevalley, the algebra P (a∗C)W is known to be polynomial

of rank r.

In the example SL(n, R), the Weyl group is given by the natural ac-tion of the permutation group Sn on the space a of traceless diagonalmatrices. Here the algebra P (a∗

C)W corresponds to the algebra of Sn -

invariants in C[T1 , . . . , Tn ]/(T1 + · · ·+Tn ), which is of course well knownto be a polynomial algebra of n− 1 generators of its own right. We notethat in the case of rank 1, the algebra D(G/K) consists of all polynomialsin the Laplace-Beltrami operator.

In the papers [12],[13], Harish-Chandra created a beautiful theory ofharmonic analysis for left K-invariant functions on the symmetric spaceG/K, culminating in a Plancherel formula for L2(G/K)K , the space ofleft-K-invariant functions on G/K, square integrable with respect to theRiemannian volume form. We will now give a brief outline of the mainresults.

For ν ∈ a∗C, we consider the following system of simultaneous eigenequa-

tions on G/K :

Df = γ(D, iν)f, (D ∈ D(G/K)). (9.2)

For r = 1, this system is equivalent to a single eigenequation for theLaplace operator. Each eigenfunction is analytic, by ellipticity of theLaplace operator. The space of K-invariant functions satisfying (9.2) isone dimensional and spanned by the so-called elementary spherical func-tion ϕν , normalized by ϕν (eK) = 1. This function can be constructedas a matrix coefficient x → 〈1K , πν (x)1K 〉, with 1K a K-fixed vectorin a suitable continuous representation of G in an infinite dimensionalHilbert space, obtained by the process of induction. By Weyl invarianceof the polynomials γ(D) it follows that ϕwν = ϕν , for all w ∈W.

In terms of the elementary spherical functions one may define theso-called Fourier transform of a function f ∈ C∞

c (G/K)K by

FG/K f(ν) =∫

G/K

f(x)ϕ−ν (x)dx, (ν ∈ a∗),

with dx the G-invariant volume measure on G/K.

52 Erik van den Ban

By analyzing the system of differential equations (9.2) it is possibleto obtain rather detailed information on the asymptotic behavior of theelementary spherical functions ϕν towards infinity. It can be shown thatthe map K/M × a → G/K, (kM,X) → k expXK is surjective. Forobvious reasons, the associated decomposition

G/K = K exp a · e (9.3)

is called the polar decomposition of G/K. In it, the a-part of an el-ement is uniquely determined modulo the action of W. Let a+ be achoice of positive Weyl chamber relative to W, then it follows thatG/K = K exp a+ · e with uniquely determined a+ -part. Moreover, themap K/M × a+ → G/K is an analytic diffeomorphism onto an opendense subset of G/K.

Accordingly, each elementary spherical function is completely deter-mined by its restriction to A+ := exp(a+). Moreover, the restrictedfunction ϕν |A+ satisfies the system of equations arising from (9.2) bytaking radial parts with respect to the polar decomposition (9.3). Usinga characterization of D(G/K) in terms of the universal algebra U(g),Harish-Chandra was able to analyze these radial differential equationsin great detail. This allowed him to show that, for generic ν ∈ C, thebehavior of the function ϕν towards infinity is described by

ϕν (k exp X) =∑

w∈W

c(wν)e(iwν−ρ)(X ) [1 + Rwν (X)], (9.4)

for k ∈ K and X ∈ a+ . Here ρ ∈ a∗ is half the sum of the positive roots,counted with multiplicities, c(ν), the so-called c-function, is a certainmeromorphic function of the parameter ν ∈ a∗

C, and Rν (X) is a certain

analytic function of X, depending meromorphically on the parameter ν.

Moreover, the asymptotic behavior of Rν is described by

Rν (tX) = O(e−tm (X )), (X ∈ a+ , t →∞), (9.5)

with m(X) a positive constant, depending on X in a locally uniformway. Each of the summands in (9.4) is an eigenfunction of the radialsystem of differential equations of its own right.

Theorem 9.1 (Harish-Chandra’s Plancherel formula). The function c

has no zeros on a∗. Moreover, let

dm(ν) :=dν

|c(ν)|2 , (9.6)

with dν a suitable normalization of Lebesgue measure on a∗ (see further


down). Then dm(ν) is Weyl-group invariant and the Fourier transformFG/K extends to an isometry from L2(G/K)K onto L2(a∗, dm(ν))W .

In footnote 3), p. 242, to the introduction of his paper [12], Harish-Chandra mentions: ‘This is reminiscent of a result of Weyl [[29], p.266] on ordinary differential equations.’ It seems that Harish-Chandrawas actually very inspired by Weyl’s paper. In [5], p. 38, A. Borelwrites: ‘[...] less obviously maybe, Weyl was also of help via his workon differential equations [29], which gave Harish-Chandra a crucial hintin his quest for an explicit form of the Plancherel measure. [...] It wasthe reading of [29] which suggested to Harish-Chandra that the measureshould be the inverse of the square modulus of a function in λ describingthe asymptotic behavior of the eigenfunctions [...] and I remember wellfrom seminar lectures and conversations that he never lost sight of thatprinciple, which is confirmed by his results in the general case as well.’

It is the purpose of the rest of this section to show that for the rankone case Theorem 9.1 is in fact a rather direct consequence of Kodaira’sgeneralization of Weyl’s result, described in Section 8.

Before we proceed it should be mentioned that in [12] and [13] Theo-rem 9.1 was completely proved for spaces of rank 1. Moreover, for thesespaces the c-function was explicitly determined as a certain quotient ofGamma factors.

For spaces of arbitrary rank Theorem 9.1 was proved modulo twoconjectures. The first of these concerned the injectivity of the Fouriertransform and the second certain estimates for the c-function. The firstconjecture was proved by Harish-Chandra himself, in his work on the so-called discrete series of representations for G, [14]. The validity of thesecond conjecture followed from the work of S. Gindikin and F. Karpele-vic, [11], where a product decomposition of the c-function in terms ofrank one c-functions was established. Simpler proofs of Theorem 9.1were later found through the contributions of [20], [10], [26].

The precise normalization of the Lebesgue measure dν may be given asfollows. The polar decomposition (9.3) gives rise to an integral formula∫

G/K

f(x) dx =∫

K

∫a+

f(k expX)J(X) dX dk, (9.7)

with dk normalized Haar measure on K, J a suitable Jacobian, anddX suitably normalized Lebesgue measure on a. The Jacobian J andthe measure dX are uniquely determined by the above formula and therequirement that J(tX) behaves asymptotically as et2ρ(X ) , for X ∈ a+

54 Erik van den Ban

and t →∞. Let dξ denote the dual Lebesgue measure on a∗. Then

dν =1|W |

dξ

(2π)n,

with |W | the number of elements of the Weyl group.We now turn to the setting of a space of rank 1. A typical example of

such a space is the n-dimensional hyperbolic space Xn , which may berealized as the submanifold of Rn+1 given by the equation x2

1 − (x22 +

· · ·+x2n ) = 1, x1 > 0. Its Riemannian metric is induced by the indefinite

standard inner product of signature (1, n) on Rn+1 . As a homogeneousspace Xn SO(1, n)/SO(n).

More generally, as a is one dimensional, all roots in R are proportional.Let α be the simple root associated with the choice of positive chambera+ . Then −α is a root as well, and possibly ±2α are roots as well. Noother multiples of α occur. We fix the unique element H ∈ a withα(H) = 1.

Via the map tH → t we identify a with R; likewise, via the maptα → t we identify a∗ with R R∗. Then a+ = ] 0,∞ [ . Rescaling theRiemannian metric if necessary we may as well assume that under theseidentifications, both dX and dξ correspond to the standard Lebesguemeasure on R.

Let m1 ,m2 denote the root multiplicities of α, 2α, i.e., mj is the di-mension of the eigenspace of ad(H) in g with eigenvalue j. Then withthe above identifications,

ρ =12(m1 + 2m2).

The Laplace operator ∆ satisfies γ(∆, iν) = (−‖ν‖2 − ‖ρ‖2) with ‖ · ‖the norm on a∗ dual to the norm on a induced by the Riemannian innerproduct on g/k p. Multiplying ∆ with a suitable negative constant,we obtain an operator L0 with γ(L0 , iν) = ν2 + ρ2 . Let L = L0 − ρ2 ,

then Lϕν = ν2ϕν .

The Jacobian J mentioned above is given by the formula

J(t) = (et − e−t)m 1 (e2t − e−2t)m 2 .

Let L := J1/2 rad (L) J−1/2 be the conjugate of the radial part of L

by multiplication with J1/2 . Put

s(ν, t) := J1/2(t)ϕν (exp tH). (9.8)

Then the system of equations (9.2) is equivalent to the single eigenequa-


tion

Ls(ν, · ) = ν2s(ν, · )

By a straightforward calculation, see [10], p. 156, it follows that

L = − d2

dt2+ q(t), q(t) =

12J−1 d2

dt2J − 1

4J−2(

d

dtJ)2 − ρ2 .

By using the Taylor series of J(t) at 0 we see that there exists a realanalytic function V on R such that

q(t) = m(m + 1)t−2 + V (t), (t > 0),

where

m =12(m1 + m2)− 1 ≥ −1

2.

On the other hand, at infinity, J(t) equals e2tρ times a power series interms of powers of e−2t with constant term 1. From this we see thatq(t) = O(e−2t), so that V (t) = O(t−2) as t → ∞. It follows that ouroperator L satisfies all requirements of Section 8.

We now observe that ϕν (e) = 1 and J(t)1/2 ∼ 2ρtm+1 (t → 0). Letc0 := 2ρ and let s1(λ, t) be defined as in Section 8, for λ ∈ C. Then itfollows that s(ν, t) = s1(ν2 , t) for all ν ∈ C. Moreover, it follows from(9.4) that the function Φν of Section 8 is given by

Φν (t) = e−tρJ(t)1/2 c(ν) eiν (1 + Rν (t)).

In particular, it depends meromorphically on the parameter ν. Fromthis it follows that the functions ν → a(ν), b(ν) are meromorphic. Byanalytic continuation it now follows that the identity (8.5) extends to anidentity of meromorphic functions. From its explicitly known form as aquotient of Gamma factors, it follows that the function ν → c(ν) has nozeros on i ] 0,∞ [ . Moreover, it has a zero of order 1 at 0. Using (8.5) wenow see that b has no zeros on i[0,∞ ] , so that the spectral measure dρ

has no discrete part. Hence,

dρ(λ) =d√

λ

2π|c(√

λ)|2

∣∣∣∣∣] 0,∞ [

.

Let F1 be the Fourier transform defined in terms of s1(λ, t) = s(√

λ, t),see Section 8. Then it follows by application of (9.7) and (9.8) that

FG/K f(ν) = F1(J1/2f exp |a+ )(ν2), (ν ∈ R), (9.9)

for every f ∈ Cc(G/K).

56 Erik van den Ban

By Corollary 8.1 the Fourier transform F1 is an isometry from thespace L2(a+ , dX) onto the space L2( ] 0,∞ [ , dρ). Moreover, the mapf → J1/2f exp |a+ is an isometry from L2(G/K)K onto L2(a+ , dX).

Finally, since W = ±I, whereas the function ν → |c(ν)|2 is even by(8.5) and (8.1), pull-back by the map ν → ν2 defines an isometry

L2( ] 0,∞ [ , dρ) −→ L2(a∗,12

dξ

2π|c(ν)|2 )W . (9.10)

By (9.9) FG/K is the composition of the three mentioned isometries.The assertion of Theorem 9.1 follows.

10 Analysis on groups and symmetric spaces

After his work on the Riemannian symmetric spaces, Harish-Chandracontinued to work on a theory of harmonic analysis for real semisim-ple Lie groups in the 1960’s. His objective was to obtain an explicitPlancherel decomposition for L2(G), the space of square integrable func-tions with respect to a fixed choice of (bi-invariant) Haar measure onG.

In the case of a compact group, the Plancherel formula is described interms of representation theory and consists of the Peter–Weyl decompo-sition combined with the Schur-orthogonality relations.

In the more general case of a real semisimple Lie group, the situationis far more complicated. If G is simple and non-compact, then the non-trivial irreducible unitary representations of G are infinite dimensional.Moreover, there is a mixture of discrete and continuous spectrum.

An irreducible unitary representation is said to be of the discrete seriesif it contributes discretely to L2(G), i.e., it is embeddable as a closedinvariant subspace for the left regular representation. Equivalently, thismeans that its matrix coefficients are square integrable. An irreducibleunitary representation has a character, which is naturally defined asa conjugation invariant distribution on G. A deep theorem of Harish-Chandra in the beginning of the 1960’s asserts that in fact all suchcharacters are locally integrable. Moreover, they are analytic on theopen dense subset of regular elements.

In [14] and [15], Harish-Chandra gave a complete classification of thediscrete series. First of all, G has discrete series if and only if it has acompact Cartan subgroup. Moreover, the representations of the discreteseries are completely determined by the restriction of their charactersto this compact Cartan subgroup. Harish-Chandra achieved their clas-


sification and established a character formula on the compact Cartanwhich shows remarkable resemblance with Weyl’s character formula.

In the early 1970’s, Harish-Chandra, [16], [17], [18], completed hiswork on the Plancherel decomposition. The orthocomplement of thediscrete part of L2(G) is decomposed in terms of representations of theso-called generalized principal series. These are induced representationsof the form

πP,ξ ,λ = IndGP (ξ ⊗ eiλ ⊗ 1),

where P is a (cuspidal) parabolic subgroup of G, with a so-called Lang-lands decomposition P = MP AP NP . Moreover, ξ is a discrete seriesrepresentation of MP and eiλ is a unitary character of the vectorialgroup AP . The space L2(G) splits into a finite orthogonal direct sum ofclosed subspaces L2(G)[P ], each summand corresponding to an equiva-lence class of parabolic subgroups with K-conjugate AP -part. Here G

counts for a parabolic subgroup, and L2(G)[G ] denotes the discrete partof L2(G).

Each summand L2(G)[P ] decomposes discretely into a countable or-thogonal direct sum of spaces L2(G)[P ],ξ parametrized by (equivalenceclasses of) discrete series representations of MP . Finally, each of thespaces L2(G)[P ],ξ has a continuous decomposition parametrized by λ ∈a∗P . Harish-Chandra achieved this continuous decomposition by reduc-tion to the space of functions transforming finitely under the action ofthe maximal compact subgroup K.

Let δL , δR be two irreducible representations of K and let L2(G)[P ],ξ ,δ

be the part of L2(G)[P ],ξ consisting of bi-K-finite functions of left K-typeδL and right K-type δR . The decomposition of this space is describedin terms of Eisenstein integrals. These are essentially K ×K-finite ma-trix coefficients of type (δL , δR ) of the induced representation involved.The Eisenstein integrals E([P ], ξ, λ, ψ) are functions on G which dependanalytically on the parameter λ ∈ a∗. In addition, they depend linearlyon a certain parameter ψ, which ranges over a certain finite dimensionalHilbert space A2(MP , ξ, δ) of functions M×K×K → C. The Eisensteinintegrals satisfy eigenequations coming from the bi-G-invariant differen-tial operators on G. As in the previous section these equations can beanalyzed in detail, and it can be shown that the integrals behave asymp-totically like

E([P ], ξ, λ, ψ)(k1 m exp X k2)

∼∑

w∈W (aQ |aP )

e(iwλ−ρQ )(X ) [cQ |P,ξ (w, λ)ψ](m, k1 , k2)

58 Erik van den Ban

for m ∈ MQ, k1 , k2 ∈ K, and as X tends to infinity in a+Q ; here Q is a

parabolic subgroup in the same equivalence class as P and W (aQ |aP )denotes the finite set of isomorphisms aP → aQ induced by the ad-joint action of K. Each coefficient cQ |P,ξ (w, λ) is an isomorphism fromthe finite dimensional Hilbert space A2(MP , ξ, δ) onto the similar spaceA2(MQ,wξ, δ). It can be shown that

cQ |P,ξ (w, λ)∗cQ |P,ξ (w, λ) = η(P, ξ, λ) I

with η(P, ξ, λ) a strictly positive scalar, independent of Q,w, δ anddepending real analytically on λ ∈ a∗P . Finally, the measure for thePlancherel decomposition of L2(G)[P ],ξ is given by

dλ

η(P, ξ, λ). (10.1)

In this sense, Weyl’s principle is valid for all continuous spectral param-eters in the Plancherel decomposition for G.

In the 1980’s and 1990’s, much progress was made in harmonic analysison general semisimple symmetric spaces. These are pseudo-Riemanniansymmetric spaces of the form G/H, with G a real semisimple Lie groupand H (an open subgroup of) the group of fixed points for an involutionσ of G. This class of spaces contains both the Riemannian symmetricspaces and the semisimple groups. Indeed the group G is a homogeneousspace for the action of G×G given by (x, y) · g = xgy−1 . The stabilizerof the identity element eG equals the diagonal H of G×G, which is thegroup of fixed points for the involution σ : (x, y) → (y, x). As a decom-position for the left times right regular action of G × G on L2(G) thePlancherel decomposition becomes multiplicity free. This is analogousto what happens for the Peter-Weyl decomposition for compact groups.

Another interesting class of semisimple symmetric spaces is formed bythe pseudo-Riemannian hyperbolic spaces SOe(p, q)/SOe(p− 1, q), p >

1.

For general semisimple symmetric spaces, M. Flensted-Jensen, [9],gave the first construction of discrete series assuming the analogue ofHarish-Chandra’s rank condition. The full classification of the discreteseries was then given by T. Oshima and T. Matsuki [25].

In [2], E.P. van den Ban and H. Schlichtkrull gave a description ofthe most continuous part of the Plancherel decomposition. Here, a newphenomenon is that the Plancherel decomposition may have finite mul-tiplicities. Nevertheless, the multiplicities can be parametrized in such away that Weyl’s principle generalizes to this context. Then, P. Delorme,


partly in collaboration with J. Carmona, determined the full Planchereldecomposition for G/H, [6], [8]. Around the same time this was alsoachieved by E.P. van den Ban and H. Schlichtkrull, [3],[4], with a com-pletely different proof. In all these works, the appropriate analogue of(10.1) goes through. For more information, we refer the reader to thesurvey articles in [1].

Parallel to the developments sketched above, G. Heckman and E. Op-dam [19] developed a theory of hypergeometric functions, generalizingthe elementary spherical functions of the Riemannian symmetric spaces.For these spaces, the algebra of radial components of invariant differen-tial operators is entirely determined by a root system and root multi-plicities. The generalization is obtained by allowing these multiplicitiesto vary in a continuous fashion. In the associated Plancherel decompo-sition, established by Opdam, [24], Weyl’s principle holds through theanalogue of (9.6).

11 Appendix: circles in P1(C)

If V is a two dimensional complex linear space, then by P(V ) we denotethe 1-dimensional projective space of lines Cv, with v ∈ V \ 0. In anatural way we will identify subsets of P(V ) with C-homogeneous subsetsof V containing 0. In particular, the empty set is identified with 0. Thegroup GL(V ) of invertible complex linear transformations of V naturallyacts on P(V ).

Let β be Hermitian form on V, i.e., β : V ×V → C is linear in the firstand conjugate linear in the second component, and β(v, w) = β(w, v)for all v, w ∈ V. By symmetry, β(v, v) ∈ R for all v ∈ V. We denote by Bthe space of Hermitian forms β on V for which the function v → β(v, v)has image R. Equivalently, this means that there exists a basis v1 , v2 ofV such that β(v1 , v1) = 1 and β(v2 , v2) = −1. It follows from this thatthe group GL(V ) acts transitively on B by g · β(v, w) = β(g−1v, g−1w).

We note that for any Hermitian form β on V the map v → β(v, · )induces a linear map from V to the conjugate linear dual space V

∗. This

map is an isomorphism if and only if β is non-degenerate. Let γ be anychoice of positive definite Hermitian form on V. Then Hβ = γ−1 β isa linear endomorphism of V ; from β(v, w) = γ(Hβ v,w) for v, w ∈ V

we see that Hβ is symmetric with respect to the inner product γ. Thecondition that β ∈ B is equivalent to the condition that Hβ has botha strictly positive and a strictly negative eigenvalue, which in turn is

60 Erik van den Ban

equivalent to the condition that det Hβ < 0. For obvious reasons we willcall B the space of Hermitian forms of signature (1, 1).

By a circle in P(V ) we mean a set of the form

Cβ := v ∈ V | β(v, v) = 0

with β ∈ B. For g ∈ GL(V ) we have g(Cβ ) = Cg ·β so that the naturalaction of GL(V ) on the collection of circles is transitive.

We now turn to the case of C2 equipped with the standard Hermitianinner product. Accordingly, any form β ∈ B is represented by a uniqueHermitian matrix H of strictly negative determinant. We will use thestandard embedding C → P1(C) := P(C2) given by z → C(z, 1). Thecomplement of the image of this embedding consists of the single point∞C := C(1, 0). The inverse map χ : P1(C) \ ∞C → C is called thestandard affine chart. It is straightforwardly verified that ∞C belongs toCβ if and only if the entry H11 equals zero. In this case the intersectionof Cβ with the standard affine chart is given by 2Re(H21z) = −H22 ,

which is the straight line −H−121 ( 1

2 H22 + iR). In particular, the form

i[z, w] = i(z1w2 − z2w1) (11.1)

is represented by the Hermitian matrix iJ (see (2.4)), and the associatedcircle in P1(C) equals the closure P1(R) := CR2 = R ∪ ∞C of the realline.

In the remaining case the circle Cβ is completely contained in thestandard affine chart, and in the affine coordinate it equals a circle withrespect to the standard Euclidean metric on C R2 . The radius r andthe center α are given by

r2 = − det H

|H11 |2, α = −H12

H11. (11.2)

The preimage under χ of the interior of this circle is the subset of P1(C)given by the inequality

sign(H11)β(z, z) < 0.

We note that all circles and straight lines in C are representable in theabove fashion. In the standard affine coordinate, the action of the groupGL(2, C) on P1(C) is represented by the action through fractional lineartransformations on C. Accordingly, we retrieve the well-known fact thatthis action preserves the set of circles and straight lines.

More generally, let v1 , v2 be a complex basis of V. Then the naturalmap z → z1v1 + z2v2 induces a diffeomorphism v : P1(C) → P(V ).


The map χv := χ v−1 : P(V ) \ Cv1 → C is said to be the affine chartdetermined by v1 , v2 . Note that z = χv (C(zv1 + v2)), for z ∈ C. Thegeneral linear group GL(V ) acts on the set of affine charts by (g, ψ) →ψ g−1 , so that g · χv = χgv . Clearly, the action is transitive. It followsthat the transition map between any pair of affine charts is given by afractional linear transformation.

From the above considerations it follows that a circle C in P(V ) cor-responds to a circle in the affine chart χv if and only if Cv1 does not lieon C. Otherwise, the circle is represented by a straight line in χv .

References[1] J.-P. Anker and B. Orsted, editors. Lie theory, volume 230 of Progressin Mathematics. Birkhäuser Boston Inc., Boston, MA, 2005. Harmonicanalysis on symmetric spaces—general Plancherel theorems.[2] E. P. van den Ban and H. Schlichtkrull. The most continuous part ofthe Plancherel decomposition for a reductive symmetric space. Ann. ofMath. (2), 145:267–364, 1997.[3] E. P. van den Ban and H. Schlichtkrull. The Plancherel decompositionfor a reductive symmetric space. I. Spherical functions. Invent. Math.,161:453–566, 2005.[4] E. P. van den Ban and H. Schlichtkrull. The Plancherel decompositionfor a reductive symmetric space. II. Representation theory. Invent. Math.,161:567–628, 2005.[5] A. Borel. Essays in the history of Lie groups and algebraic groups,volume 21 of History of Mathematics. American Mathematical Society,Providence, RI, 2001.[6] J. Carmona and P. Delorme. Transformation de Fourier sur l’espacede Schwartz d’un espace symétrique réductif. Invent. Math., 134:59–99,1998.[7] E. A. Coddington and N. Levinson. Theory of ordinary differentialequations. McGraw-Hill Book Company, Inc., New York-Toronto-London,1955.[8] P. Delorme. Formule de Plancherel pour les espaces symétriques ré-ductifs. Ann. of Math. (2), 147:417–452, 1998.[9] M. Flensted-Jensen. Discrete series for semisimple symmetric spaces.Ann. of Math. (2), 111:253–311, 1980.[10] R. Gangolli. On the Plancherel formula and the Paley-Wiener theo-rem for spherical functions on semisimple Lie groups. Ann. of Math. (2),93:150–165, 1971.[11] S. G. Gindikin and F. I. Karpelevič. Plancherel measure for symmetricRiemannian spaces of non-positive curvature. Dokl. Akad. Nauk SSSR,145:252–255, 1962.[12] Harish-Chandra. Spherical functions on a semisimple Lie group. I.Amer. J. Math., 80:241–310, 1958.[13] Harish-Chandra. Spherical functions on a semisimple Lie group. II.Amer. J. Math., 80:553–613, 1958.

62 Erik van den Ban

[14] Harish-Chandra. Discrete series for semisimple Lie groups. I. Con-struction of invariant eigendistributions. Acta Math., 113:241–318, 1965.[15] Harish-Chandra. Discrete series for semisimple Lie groups. II. Explicitdetermination of the characters. Acta Math., 116:1–111, 1966.[16] Harish-Chandra. Harmonic analysis on real reductive groups. I. Thetheory of the constant term. J. Functional Analysis, 19:104–204, 1975.[17] Harish-Chandra. Harmonic analysis on real reductive groups. II.Wavepackets in the Schwartz space. Invent. Math., 36:1–55, 1976.[18] Harish-Chandra. Harmonic analysis on real reductive groups. III.The Maass-Selberg relations and the Plancherel formula. Ann. of Math.(2), 104:117–201, 1976.[19] G. J. Heckman and E. M. Opdam. Root systems and hypergeometricfunctions. I. Compositio Math., 64:329–352, 1987.[20] S. Helgason. An analogue of the Paley-Wiener theorem for the Fouriertransform on certain symmetric spaces. Math. Ann., 165:297–308, 1966.[21] D. Hilbert. Grundzüge einer allgemeinen Theorie der linearen Inte-gralgleichungen. Chelsea Publishing Company, New York, N.Y., 1953.[22] I.S. Kac. The existence of spectral functions of generalized secondorder differential systems with boundary conditions at the singular end.Amer. Math. Soc. Transl. (2), 62:204–262, 1964.[23] K. Kodaira. The eigenvalue problem for ordinary differential equa-tions of the second order and Heisenberg’s theory of S-matrices. Amer.J. Math., 71:921–945, 1949.[24] E. M. Opdam. Harmonic analysis for certain representations of gradedHecke algebras. Acta Math., 175:75–121, 1995.[25] T. Oshima and T. Matsuki. A description of discrete series forsemisimple symmetric spaces. In Group representations and systems ofdifferential equations (Tokyo, 1982), volume 4 of Adv. Stud. Pure Math.,pages 331–390. North-Holland, Amsterdam, 1984.[26] J. Rosenberg. A quick proof of Harish-Chandra’s Plancherel theoremfor spherical functions on a semisimple Lie group. Proc. Amer. Math.Soc., 63:143–149, 1977.[27] E. C. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Oxford, at the Clarendon Press, 1946.[28] H. Weyl. Singuläre Integralgleichungen. Math. Ann., 66:273–324,1908.[29] H. Weyl. Über gewöhnliche Differentialgleichungen mit Singularitätenund die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann.,68:220–269, 1910.[30] H. Weyl. Über gewöhnliche lineare Differentialgleichungen mitsingulären Stellen und ihre Eigenfunktionen. (2. Note). GöttingerNachrichten, pp. 442–467, 1910.[31] H. Weyl. Ramifications, old and new, of the eigenvalue problem. Bull.Amer. Math. Soc., 56:115–139, 1950.

3Weyl’s Work on

Singular Sturm–Liouville OperatorsW.N. Everitt

School of Mathematics and StatisticsUniversity of Birmingham

[email protected]

H. KalfMathematisches Institut

Universität Mü[email protected]

Wie doch ein einziger Reicher so viele Bettler in NahrungSetzt! Wenn die Könige baun, haben die Kärrner zu tun.

(Schiller, Kant und seine Ausleger)

Abstract

Up to the year 1910 there had been many significant mathematical con-tributions to the theory of linear ordinary differential, and of linear in-tegral equations. Many of these advances were based on the originalstudies initiated by Sturm and Liouville commencing in 1829. In theclosing years of the 19th century the work lead by the Göttingen schoolof mathematics gave a much needed overview of these significant andvaried contributions to mathematical analysis.

The contributions of Hermann Weyl, in and around the year 1910,to the theory of Sturm-Liouville theory heralded the modern analyti-cal and spectral study of boundary value problems. In particular thepaper written for Mathematischen Annalen in 1910 stands today as alandmark not only in Sturm-Liouville theory, but in the development ofmathematical analysis in the 20th century.

This paper discusses the work of Weyl, and indeed of the Göttin-gen school of mathematics, in introducing the now familiar terms ofSturm-Liouville theory; limit-point and limit-circle endpoint classifica-tions; the point, continuous and essential spectra; singular eigenfunctionexpansions; and the interplay of these results with the development ofquantum theory in physics.

63

64 W.N. Everitt and H. Kalf

1 Introduction

The investigation of second-order linear ordinary differential equationshas a long and fascinating history, extending back to the middle of the18th century. It shaped the concept of a function, led to Cantor’s settheory and influenced the theories of measure and integration. It wasessential to solving the initial-boundary-value problems for partial differ-ential equations, for example the heat and wave equations, by separationof the variables. In particular, the study of the expansion of a functionin terms of eigenfunctions of Sturm-Liouville equations affected the the-ory of integral equations and the early shape of functional analysis, andwas in turn influenced by these disciplines.

When Hilbert in 1906 in the 4th of the six communications of the out-line of a general theory of linear integral equations, introduced the newconcept of a continuous spectrum (Streckenspektrum [25, Page 122]) itwas natural to try to consider expansion theorems similar to the Fourierintegral theorem, in the light of this new notion. The first mathematicianto undertake this task was Hilb in his Erlangen Habilitationsschrift [22],who had spent the summer semester of 1906 in Göttingen, but it wasthree papers by Weyl, written in the years 1908 to 1910, that clinchedthe issue, see [55], [56] and [57]. The most detailed of these papers is[56]1 , which was submitted in 1909 and was the basis of his Habilitationthe following year, and it is this memoir and its influence with which weare primarily concerned in this present paper. In [58] Weyl illustratedhis theory by means of the example of the Fourier transform on theinterval (−∞,∞). The difficulty with this seemingly simplest possiblecase is that both endpoints of the interval are singular and that the mul-tiplicity of the spectrum is two. Weyl had extended the results in [56] tothis situation in [57], giving as interesting examples the Bessel and hy-pergeometric differential equations, the Hankel transform yielding theexpansion theorem in the former case. (The Fourier expansion in anL2(R)-setting is summarised by the Plancherel theorem. Replacementsof the real line by spaces with a more complicated group structure, as byHarish-Chandra, are discussed by van den Ban in this present volume,see [3].)

In 1935 Weyl returned to his Habilitationsschift [56], albeit very briefly(see the references to [56] in Sections 2 and 7 below), when he consideredin [59] a problem which is connected with the difference analogue of the

1 A translation into English of the first three of the four chapters can be found in[37].

Weyl’s work on singular Sturm–Liouville operators 65

Sturm-Liouville equation that had previously been treated by Hellinger[20]. Weyl’s Gibbs lecture of 1948 [61] is a tour d’horizon of the diverseproblems he pressed his mark upon, in particular putting the results of[56] into a historical perspective.

Extensive material on Weyl, biographical and otherwise, is mentionedon the web site of the MacTutor History of Mathematics, so that there isno need to give such details here in this present paper. The most detailedsource of information on Hilb is to be found in [52]. Illuminating generalaccounts of the historical development of spectral theory are given in [5]and [10] (see also the epilogue in [26]). For the development of Sturm-Liouville theory we refer to the introduction in [1] and more specificallyto [14]. The references [11, Pages 1581-92], [12, Chapter 2] and [13] arealso of interest.

2 The Weyl circle method

For continuous q : [0,∞) → R and p : [0,∞) → (0,∞) Weyl in [56]considers the Sturm-Liouville differential expression

L(u) := −(pu′)′ + qu, (2.1)

emphasising that in contrast to previous contributions (including hisown paper [55]) the coefficient p need not be continuously differentiable,i.e., he considers functions u ∈ C1([0,∞)) such that pu′, which wouldmuch later be called the quasi-derivative of u, is again in C1([0,∞)). Forsuch functions u, v the Lagrange identity reads∫ x

0(L(u)v − uL(v)) =

∫ x

0

(upv′ − pu′v

)′= [u, v](x)− [u, v](0), (2.2)

where

[u, v](x) = u(x)(pv′)(x)− (pu′)(x)v(x).

When the coefficient p = 1 the term [u, v] is the Wronskian of u and v.

It is natural to impose a boundary condition at 0 of the form

R1(u) := u(0) cos(α) + (pu′)(0) sin(α) = 0, (2.3)

where α ∈ [0, π). The question is: What kind of boundary condition, ifany, does one have to impose at infinity in order to obtain a properlyformulated eigenvalue problem and an associated expansion formula?


For any λ ∈ C the equation2

L(u) = λu (2.4)

has a fundamental system of solutions U1(·, λ), U2(·, λ) satisfying theinitial conditions

U1(0, λ) = 1 (pU ′1)(0, λ) = 0

U2(0, λ) = 0 (pU ′2)(0, λ) = 1.

For fixed x ≥ 0, the solutions U1(x, ·), U2(x, ·) are entire functions on C.

Let

u1 := U1cos(α) + U2 sin(α) and u2 := −U1 sin(α) + U2cos(α). (2.5)

Then u2 satisfies the boundary condition (2.3). Taking λ ∈ C \ R andβ ∈ [0, π), Weyl asked: For which w ∈ C does u1 + wu2 satisfy a realboundary condition at a > 0

R2(u) := u(a)cos(β) + (pu′)(a) sin(β) = 0 ? (2.6)

Now R2(u2) = 0, because otherwise the non-real number λ would be aneigenvalue of the symmetric problem (2.3), (2.4), (2.6). Hence

w = −R2(u1)R2(u2)

= −u1(a, λ) cos(β) + (pu′1)(a, λ) sin(β)

u2(a, λ) cos(β) + (pu′2)(a, λ) sin(β)

or, defining h := cot(β),

w = l(h) := −Ah + B

Ch + D. (2.7)

In view of

AD −BC = [u1 , u2 ](0) = 1, (2.8)

the fractional linear transformation (2.7) is non-degenerate. Since itsdenominator never vanishes, l(·) maps the extended real line onto acircle, Ca(λ), the equation of which is straightforward to determine.

On account of (2.2) and (2.4) we have

2i Im(λ)∫ a

0|u1 + wu2 |2 = [u1+wu2 , u1+wu2 ](a)−[u1+wu2 , u1+wu2 ](0)

(2.9)for all w ∈ C. Now

[u1 , u1 ](0) = 0 = [u2 , u2 ](0) (2.10)

2 There is no problem to replace λ by λk as long as the weight function k : [0,∞) →R is positive. The case that k may change sign is briefly considered in [56].


and (2.8) imply

[u1 + wu2 , u1 + wu2 ](0) = −2i Im(w).

Since the two terms

u1(a, λ) + wu2(a, λ) and (pu′1)(a, λ) + w(pu′

2)(a, λ)

are linearly dependent if and only if w ∈ Ca(λ), we have

[u1 + wu2 , u1 + wu2 ](a) = 0 (2.11)

if and only if w ∈ Ca(λ) and so

Ca(λ) =

w ∈ C :∫ a

0|u1(x, λ) + wu2(x, λ)|2 dx =

Im(w)Im(λ)

, (2.12)

i.e., if Im(λ) > 0 ( Im(λ) < 0 ) the set of w ∈ C for which u1 + wu2

satisfies a real boundary condition at a is a circle in the upper (lower)complex half-plane C+ ( C− ). With w = 0 certainly lying in the exteriorof Ca(λ), the interior of Ca(λ), the disc Da(λ), is obtained by replacing“ = ” with “ < ” in (2.12). Hence

Da2 (λ) Da1 (λ) for 0 < a1 < a2 < ∞.

As a → ∞ the nesting circles Ca(λ) shrink either to a circle (this iscalled the limit-circle case, LCC) or to a point (named the limit-pointcase, LPC) [56, Satz 1]. In any case it follows from (2.12) that for everyλ ∈ C \ R equation (2.4) has at least one non-trivial solution in L2(0,∞),viz., u1 +wu2 when w ∈ C∞(λ) [56, Satz 2]. Jumping ahead of our story,the operator-theoretic version of this result is the following. If L is themaximal operator associated with (2.1) in H := L2(0,∞) (see equation(7.1) below), the null space of L − λ is non-trivial for every λ ∈ C \ R.

The operator-theoretic proof is as follows. Suppose N(L− λ) = 0 forsome λ ∈ C \ R. Then (7.2) implies

0 = N(L∗0 − λ) = R(L0 − λ)⊥.

This result is also true with λ replaced by λ, because the coefficients of(2.1) are real-valued. As a consequence

R(L0 − λ) = H = R(L0 − λ),

i.e., L0 and so L are self-adjoint operators in H. However, L fails to besymmetric since

〈Lu1 , u2〉 − 〈u1 , Lu2〉 = −[u1 , u2 ](0) = −1


by (2.8), where u1 , u2 are smooth functions with compact support whichcoincide with u1 , u2 in a right neighbourhood of 0.

In order to determine the radius ra(λ) of Ca(λ) we proceed as in[48, Section 2] and observe that the centre za(λ) and ∞ = l(−D/C)are mirror points with respect to the circle. Therefore l−1(za(λ)) andl−1(∞) are mirror points with respect to the real line, i.e.,

l−1(za(λ)) = l−1(∞) = −D

C

or

za(λ) = l(−D/C) = −AD −BC

CD − CD.

Since l(0) is certainly a point on Ca(λ), we havera(λ) = |l(0)− za(λ)| =

∣∣∣∣AD −BC

CD − CD

∣∣∣∣=

1|[u1 , u2 ](a)|

=(2 |Im(λ)|

∫ a

0 |u2(x, λ)|2 dx)−1

,

(2.13)using (2.8) and (2.2) together with (2.10). Both in [56] and [61] Weylhimself determined ra(λ) by observing that the straight lines definedby the real and imaginary parts of R2(u1) + wR2(u2) are orthogonal toeach other, although he does use a fractional linear transformation in[61]. Rellich in his presentation of the Weyl theory, see [41] and [27],chose to avoid properties of fractional linear transformations altogetherand was followed in this respect by Hellwig [21].

An important consequence of (2.13) is that u2 is also of integrablesquare in the LCC. So all solutions of (2.4) then have this property. NextWeyl showed that if all solutions of (2.4) are in L2(0,∞) for some λ0 thenthis is also true for all other values of λ [56, Satz 5]. By the time he wrote[59] he regarded his original proof via an integral equation as unnatural.His argument in [59, Page 240 ff.], when adopted to the Sturm-Liouvillecase, amounts to applying the variation-of-constants formula to

(L− λ0)u = (λ− λ0)u =: f

(u is a solution of (2.4)), which is the standard proof used today ([41,Page 31 ff.], [27, Page 125 ff.], [21, Page 224 ff.]). Weyl found the issueimportant enough to append a note to Satz 5 when [56] and [59] werereprinted in 1955 in his Selected Papers. This note also went into his


Collected Papers, but unfortunately without adjusting the page refer-ences to the new situation.

The operator-theoretic background of this result is that the defectnumber

dλ := dimR(L0 − λ)⊥

of a symmetric operator in a Hilbert space is a constant in C+ and inC− (in addition dλ = dλ , since the coefficients of (2.1) are real-valued).Here we have the slightly more detailed information that dλ = 2 for allλ ∈ C in the LCC. The LPC is of course also independent of λ ∈ C andin this case we have dλ = 1 if λ ∈ C \ R.

3 Boundary conditions at infinity

Let w ∈ C∞(λ) and x ≥ 0. Abbreviate

uw (x) := u1(x, λ) + wu2(x, λ), (3.1)

and then define a Green’s function by

Gw (x, y;λ) :=

u2(x, λ)uw (y) if 0 ≤ x ≤ y <∞u2(y, λ)uw (x) if 0 ≤ y < x < ∞.

(3.2)

For continuous f ∈ L2(0,∞) let

v(x) :=∫ ∞

0Gw (x, y;λ)f(y) dy = uw (x)g(x) + u2(x, λ)hw (x),

where

g(x) :=∫ x

0u2(y, λ)f(y) dy and hw (x) :=

∫ ∞

x

uw (y)f(y) dy.

Let ai be a sequence of real numbers which tends to infinity and letw ∈ C∞(λ). Take a sequence wi with wi ∈ Cai

(λ) which converges tow in C. Using (2.11) as well as (2.2) and (2.10), we see that

[v, uwi] (ai) = g(ai) [uwi

, uwi] (ai) + hwi

(ai) [u2 , uwi] (ai)

= hwi(ai)2i Im(λ)

∫ ai

0u2uwi

.

In the LCC this last expression tends to zero as i → ∞. From thisresult Weyl concludes in the LCC a one-parameter family of boundaryconditions has to be imposed on the functions in

ϑ :=u ∈ C1([0,∞)) : pu′ ∈ C1([0,∞) and u,L(u) ∈ L2(0,∞)

(3.3)


in addition to (2.3), viz.,

lima→∞

[u, u1 + wu2 ](a) = 0, (3.4)

with w being any point on the circle C∞(λ). In the LPC the Green’sfunction (3.2) is completely determined by w ∈ C∞(λ) and no bound-ary condition, in addition to (2.3), has to be imposed on the functionsin ϑ given by (3.3); the choice of λ ∈ C \ R is irrelevant. Using vonNeumann’s theory of defect numbers it can in fact be shown that in theLPC condition (3.4) is automatically satisfied for every u ∈ ϑ (see, e.g.,[53, Satz 13.19a]).

Weyl then shows that the LPC occurs when

q(x) ≥ −c (x ≥ 0)

for some c ≥ 0. Indeed, choose λ := −(c + 1) and consider the functionu := U1(·, λ); then u is positive in a neighbourhood to the right of 0. Ata first zero x0 > 0 (zeros of non-trivial solutions of (2.4) are isolated) wewould have u′(x0) < 0, which is incompatible with

(pu′)(x) =∫ x

0(pu′)′ =

∫ x

0(q − λ)u ≥

∫ x

0u (x ∈ [0, x0 ]).

Hence u is strictly increasing and so not in L2(0,∞). Weyl himself relieson a representation of u to yield this result, but arguments of the typejust given are familiar in oscillation theory and are employed by him toprove the results mentioned at the beginning of Section 6 below.

The fact that there is no growth restriction on p is particularly strikingin the light of the multidimensional analogue of (2.1),

L(u) := −n∑

i,k=1

∂i(aik∂k ) + q,

where the aik are the continuously differentiable entries of a strictly pos-itive matrix function A. In general, L does not have a unique self-adjointrealisation in L2(Rn ), even if q is absent, unless there is some growthrestriction on the largest eigenvalue of A (see [8] and the literature citedtherein).

4 The spectrum in the limit-circle case

In the LCC it follows from (3.2) that∫ ∞

0

(∫ ∞

0|Gw (x, y;λ)|2 dy

)dx <∞


for all w ∈ C∞(λ). As a consequence the resolvent of every operator S

with domain (3.3) and boundary conditions (2.3) and (3.4), is a com-pact operator (completely continuous [vollstetig] in the terminology ofthe time; in fact it belongs to a very special class of compact operators,the Hilbert-Schmidt operators), so that Hilbert’s results for such opera-tors are applicable. The spectrum of S is purely discrete; the eigenvalues- they are all simple in our case - have no finite accumulation point. Thepoint +∞ is always an accumulation point, but −∞ may be an accumu-lation point as well (this is not explicitly mentioned in [56], though). Thecorresponding eigenfunctions ϕi form a complete orthonormal systemin the sense that for f ∈ D(S) we have

f =∑

i

〈f, ϕi〉ϕi

where∑

i |〈f, ϕi〉ϕi | is uniformly convergent on every compact subset of[0,∞) [56, Satz 4]. So the LCC is very similar to the regular case whenp > 0 and q are continuous on a compact interval (in which case thereare, however, at most finitely many negative eigenvalues). We add tothese results that there is norm convergence if f ∈ L2(0,∞).

5 Invariance of the essential spectrum

Using what is now called Weyl sequences or singular sequences, Weylhad shown in [54] that the union of the continuous spectrum and theaccumulation points of the point spectrum3 is invariant under compactperturbations. Assuming that there is LPC at ∞ (otherwise the essen-tial spectrum is empty, as we have just seen), Weyl remarks that thedifference of the two resolvents corresponding to a boundary conditionof the form (2.3) with α ∈ [0, π) and γ ∈ [0, π), γ = α, is a compactoperator and so the essential spectrum is independent of the boundarycondition [56, Satz 8]. The simple proof is as follows. We give the func-tions in (2.5) and (3.1) the indices α and γ, denoting by wα,wγ thecorresponding limit points. Since we are in the LPC, there is a numberC = 0 such that

uwα(x) = Cuwγ

(x) (x ≥ 0).

It suffices to find c1 , c2 ∈ C with

c1u2α − c2u2γ = uwα, (5.1)

3 The name “essential spectrum” for this union is due to Wintner [62].


because then

c1C−1u2α (x, λ)uwα

(y)− c2u2γ (x, λ)uwγ(y) = uwα

(x)uwγ(y) (x, y ≥ 0).

We note that (5.1) is equivalent to

c2 sin(γ)− c1 sin(α) = cos(α)− wα sin(α)

−c2 cos(γ) + c1 cos(α) = sin(α) + wα cos(α),

which has a non-zero determinant. In operator-theoretic language Weyl’sresult can be rephrased as follows: Varying the boundary conditions ofa Sturm-Liouville operator is a very special compact perturbation, viz .,a rank-one perturbation (see, e.g., [53, Satz 10.17].

Weyl emphasises that he has shown in [54] that the continuous spec-trum of a bounded (self-adjoint) operator is in general not invariantunder compact perturbations and says, “At first glance it is plausible toconjecture that the continuous spectrum itself remains unaltered whenthe boundary condition is changed, but I am unable to confirm this”. In1957 Aronszajn, using the inverse spectral theory of Gel’fand-Levitan,produced a counter-example [2].

A more explicit and particularly amazing example is

p(x) = 1 and q(x) = cos(√

x)

(x ≥ 0)

for which the essential spectrum is [−1,∞). The absolutely continuousspectrum is (1,∞), and this part of the spectrum is independent of theboundary condition by virtue of a result of Aronszajn in [2]. For almostall α ∈ [0, π) there is a dense point spectrum in [−1, 1], but for a denseset of angles in [0, α) the spectrum is singular continuous in [−1, 1].Further explanations and references can be found in [9].

6 The spectrum in the limit-point case

Weyl shows that in the LPC there is an expansion formula which consistsof two terms, a Fourier series which takes care of the (finite or infinite,possibly empty) set of eigenvalues and a Riemann-Stieltjes integral theintegrand of which involves the function u2 (which satisfies the bound-ary condition (2.3)) while the integrator, responsible for the continuousspectrum, is built up from solutions of (2.4) in a complicated way. Theproof of this result [56, Satz 7], which Weyl regarded as the principalaim of his paper, is a veritable tour de force, using in particular, fromhis 1908 dissertation, his extension of Hilbert’s theory to integral ker-nels on unbounded intervals, and from Hellinger’s 1907 dissertation the


concept of “eigendifferentials”. A certain drawback of this form of theexpansion theorem is mentioned in Section 8 below. However, Weyl’sidea to relate the asymptotic behaviour of the solutions of (2.4) to thecontinuous part of the spectral function led him to significant results insituations similar to (6.1), (6.2) and (6.4) below.

Extending Sturm’s classical results, Weyl proves that the spectrum ispurely discrete if

limx→∞

q(x) = ∞.

More generally, if c := lim infx→∞ q(x) is finite, then the spectrum ispurely discrete below c, the number of eigenvalues (they are all simple)being equal to the number of zeros in (0,∞) of the solution (unique upto a factor) of (2.3) and (2.4), with λ = c. Arranging the eigenvaluesin a natural way, the eigenfunction belonging to the n-th eigenvalue hasexactly n− 1 zeros in (0,∞) [56, Satz 9].

To describe his results concerning the continuous spectrum we assumefor simplicity that p = 1 on [0,∞). Suppose∫ ∞

0|q(x)| dx < ∞ (6.1)

or

∫ ∞

0x |q(x)| dx <∞. (6.2)

Weyl proves that there are at most finitely many negative eigenvalueswhen (6.2) holds. For λ > 0 he shows under condition (6.1) that thesolutions of (2.4) behave asymptotically like sin(x

√λ) and cos(x

√λ) for

large x and fixed λ and, also assuming (6.2), he is able to write thespectral function as

ρ(λ) =1π

∫ λ

0

1√s[m2

1(s) + m22(s)]

ds (λ > 0), (6.3)

where the functions m1 and m2 are related to the asymptotic behaviourof the solution u2 . It was half a century later that Weyl’s conditions(6.1) and (6.2) and his technique became prominent in the work of L.D.Fadeev on inverse scattering (see [33]).

In the case

p(x) = 1 and q(x) = −x (x ≥ 0) (6.4)

the solutions are Bessel functions of order ±1/3. From their known


asymptotic behaviour Weyl derives the result that the spectral functionis continuously differentiable on the whole real line.

7 Interplay with quantum mechanics

In his obituary for Hilbert, Weyl mused on the timeliness of discoveriesand said [60, Page 645], “Most scientific discoveries are made when ‘theirtime is fulfilled’; sometimes, but seldom, a genius lifts the veil decadesearlier than could have been expected”. Precisely this had happenedwith Weyl’s papers [55], [56] and [57]; there was a full-fledged spectralanalysis of one-dimensional or spherically symmetric quantum mechan-ical systems about 16 years before wave mechanics was discovered bySchrödinger.

This situation was acknowledged by Fues, Schrödinger’s assistant inZurich at the time, when he wrote [17, Page 295], “Prof. H. Weyl kindlyprovided me with papers of his own from which everything emerges thatis necessary from the point of view of wave mechanics”. It is well knownthat Schrödinger himself contacted his colleague in connection with hisequation. There is an acknowledgement at the beginning of the first ofhis four communications “Quantisation as an eigenvalue problem”, andin the fourth, referring to the difficulties the presence of the continuousspectrum causes, he writes, “The theory of such integral representationswas developed by H. Weyl, however for ordinary differential equationsonly, but a generalisation to partial differential equations should proba-bly be permitted.” [46, Page 124]. In a footnote he adds, “I am indebtedto Mr H. Weyl not only for these references to the literature4 , but alsofor his very valuable oral instructions concerning these not very easythings.”.

These things were and are indeed not very easy and Kemble in hiscarefully written early book on quantum mechanics states that [56] is“a basic paper which unfortunately involves an elaborate mathematicaltechnique and makes difficult reading for the non-specialist”, [30, Page163]. Since it suggests itself to regard the Fourier integral as a lim-iting case of a Fourier series, he outlined heuristically in four starredsections [30, Pages 165 to 173] a transition from the regular case toa general expansion theorem. Mathematically, this approximation byregular problems was realised around 1950 independently by Levitan,

4 Schrödinger mentions [55], [56] and two papers by Hilb - not the ones in our listof references.


Levinson and Yosida. The simplest account is probably that given inthe book by Coddington and Levinson [7, Chapter 9].

The development of quantum mechanics had enormous repercussionson mathematics. In analysis, notably through the work of John vonNeumann and Stone, it led to a theory of unbounded operators in anabstract Hilbert space, culminating in von Neumann’s spectral theoremfor self-adjoint operators. Weyl himself turned to the group-theoreticalproblems the new physical theory posed. This story has frequently beentold, see [5] and [10]. Here we restrict ourselves to more technical details,in connection with Weyl’s paper [56].

The abstract framework demands or is facilitated by the use of closedoperators in a complete space.. This was why Stone, under now thecommon assumptions (I is an open interval)

p, q : I → R with p−1 , q ∈ L1loc(I),

associated with (2.1) a maximal operator L in the Hilbert space H :=L2(I) with domain

D(L) := u ∈ ACloc(I) : pu′ ∈ ACloc(I) and u,L(u) ∈ H, (7.1)

relaxing the requirement on a function to be continuously differentiableas in (3.3), to that of being locally absolutely continuous. In addition itis convenient to consider the smaller space

ϑ0 := u ∈ D(L) : supp(u) ⊂ I is compact.

L0 , the closure of L ϑ0 , is called the minimal operator associated with(2.1), and Stone shows that

L∗0 = L (7.2)

[47, Pages 459 onwards]. Sometimes, as in questions of oscillation prop-erties, it is necessary to demand p > 0 almost everywhere on I, but oth-erwise the analysis given in Sections 2 to 6 above is in no essential wayaffected by the new generalisation5 . Stone shows that the self-adjoint re-alisations of (2.1) with separated boundary conditions are exactly thosewhere u ∈ D(L) is restricted by (3.4) ([47, Page 457 onwards]; more pre-cisely he does this for slightly different spaces which are in one-to-onecorrespondence with (3.4) and which involve the function u2 only). A

5 Weyl wrote in [59, 649], “I think Hilbert was wise to keep within the bounds ofcontinuous functions when there was no actual need for introducing Lebesgue’sconcepts.”


direct proof of this result was later given by Rellich [41] and simplifiedby A. Schneider [45] (see also [21, Pages 239-243]).

The boundary conditions (3.4) (or those of Stone), while satisfactoryfrom a theoretical point of view, are often not easy to apply to specificproblems. In case p and q are such that the singular point of (2.4),here at ∞, is a regular singular point in the sense of complex analysis,Rellich [40] was able to replace (3.4) by a boundary condition that looksexactly like a regular boundary condition, but where u(∞) and (pu′)(∞)(which in general do not exist) are replaced by “initial numbers” (seealso [21, Chapter 15]). A modification of the Weyl circle method and of(3.4), which is inspired by the idea that problems which arise entirelywithin the realm of the real line should also be described by boundaryconditions which involve real-valued functions only, is due to Mohr [34](see also [29]). He illustrated the efficiency of his method in a number ofpapers; we only mention [35] here where he supplements (6.1) and (6.2)by conditions on p and the weight k which are more natural than thoseimposed by Weyl in [56] and [57].

Since the minimal operator has a one-parameter family of self-adjointextensions in the LCC, it is natural to ask whether there is one exten-sion which is distinguished from a mathematical or physical point ofview. The first to raise and answer this question was Friedrichs [15] and[16]. He once said that all he used mathematically he obtained fromWeyl - “except for Hilbert space and that I got from von Neumann” [39].Friedrichs showed that such a distinguished extension exists when theminimal operator is bounded from below, but it took some time beforea description of his extension was given by Rellich [42] in terms of aparticular fundamental system of solutions of (2.4). In respect of laterwork on the Friedrichs extension generated by Sturm-Liouville differen-tial expressions, see the papers of Rosenberger [44], and of Niessen andZettl [37].

8 The expansion theorem and the m-function

Weyl’s derivation of the expansion theorem was criticised by Hilb as“rather complicated” [23, Page 334] and his expansion formula as “notvery transparent” [24, Page 1265]. He writes in italics [23, Page 334],“We want to show how easy the desired expansions can be obtainedwithout this theory [Weyl’s use of singular integral equations] directlyfrom Cauchy’s residue theorem”. To do this, he needs to know thatthe Green’s function is analytic on C+ in the spectral parameter. Let


λ0 , λ ∈ C+ and w0 , w = m(λ) the corresponding Weyl limit points (weassume that there is LPC at ∞). Hilb then observes that Hilbert’sresolvent identity [25, Page 140]

Gw 0 (x, y;λ0)

= Gw (x, y;λ)− (λ− λ0)∫ ∞

0Gw 0 (x, z;λ0)Gw (z, y;λ) dz (x, y ≥ 0)

can be viewed as an integral equation for Gw . This integral equationcan be solved by means of a Neumann series, which is a power series inλ−λ0 . (The radius of convergence can be shown to be Im(λ0).) It followsin particular that w as a function of λ - which is what in the literature iscalled the Titchmarsh-Weyl, Weyl-Titchmarsh or even Weyl m-functionis analytic on C+ [23, Page 335]. (While Hellinger [20] applied the Vitalitheorem to establish the analyticity of the m-coefficient for the differenceanalogue of (2.4), Weyl used Hilb’s argument in [59, Page 244]6 ; it occursagain in [61, Page 120].) Hilb then derived a formula which A. Knesercalled the Fourier-Hilb integral representation, and which he evaluatedin the Fourier and Bessel cases [31, Sections 51 and 52]. However, Weyllater remarked [61, Page 124] that Hilb did not carry the analysis “so faras to obtain the explicit construction of the differential dρ [the spectralmeasure]”.

However, once it is known that the limit point w = m(λ) is an analyticfunction, connection with the spectral function ρ can be made as follows.Performing the limit a →∞ in (2.12), it follows that

Im(m(λ)) > 0 if Im(λ) > 0.

Such analytic functions (they are called “positive” by Weyl [59, Page231], but today the names of Herglotz or Nevanlinna are more frequentlyattached to them) can be represented in the form

m(λ) = c1λ + c2 +∫ ∞

−∞

(1

s− λ− s

s2 + 1

)dρ(s) (8.1)

where ρ : R → R is non-decreasing, and satisfies the growth condition∫ ∞

−∞

11 + s2 dρ(s) <∞;

then in (8.1)

c1 := limt→∞

Im(m(it))t

and c2 := Re(m(i)).

6 We note in passing that this is Kodaira’s reference for the analyticity of m [32,Theorem 1.2].


(The representation (8.1) is unique if ρ is required, say, to be continuousfrom the right and normalised by ρ(0) = 0.)

The study of analytic functions f on the unit disc which have a positivereal part was initiated by Carathéodory in 1907. F. Riesz showed in 1911that such functions can be represented (essentially uniquely) in the form

f(z) = Im(f(0)) +∫ 2π

0

exp(it) + z

exp(it)− zdω(t) (|z| < 1) (8.2)

where ω is a non-decreasing function. In the same year a simplified proofwas given by Herglotz, and so it is widely believed that the representation(8.2) originates with him (see, e.g., [47, Page 571]). The representation(8.1) follows from (8.2) after a conformal mapping. Now the Stieltjesinversion formula applied to (8.1) yields

ρ(λ) = limε0

1π

∫ λ

0Im(m(s + iε)) ds (λ ∈ R), (8.3)

which is the desired connection.Formula (8.3) was first derived by Titchmarsh in the 5th of his se-

ries of eight papers on expansions in eigenfunctions ([49, (4.3)]; that hisfunction k is indeed up to a factor π the spectral function ρ follows from[49, (5.1) to (5.3)]. He did not seem to be aware of formula (8.1) but hewas certainly aware of the representation formula (8.2), because he citedChapter VII, Section 2, of Nevanlinna’s book [36], where (8.2) is provedand called the Poisson-Stieltjes representation. Titchmarsh, however,does not make the transformation to C+; rather he prefers to give aseparate proof of (8.3), taking the Cauchy integral for the resolvent ashis starting point7 . In [48, Page 40] he had established the analyticity ofthe m-coefficient without knowing of Hilb’s paper [23]. It is only the 2nd

edition of his book [51, Page 188] which has a reference to [23]. In [50,Section 6] he proves that the spectral function is continuously differen-tiable on (0,∞) under the condition (6.1), showing that the integrandin (6.3) is Im(m(s)) (see also [51, Page 118]).

Formula (8.3) was derived independently, though slightly later, byKodaira [32] in a way that Weyl found to be more direct, [61, Page 124]and [62, Page 168]. Stone had shown that the projection operators whichform the spectral resolution of a general self-adjoint operator are, in our

7 Mary Cartwright concludes her obituary [6] of Titchmarsh in the following way. “Itseems that his antipathy to geometry prevented him from using certain methodswhich would have led to the kind of simplification at which he aimed. His ownsimplifications paved the way for others to achieve further improvements, but itmay be that, after all, it will be for his results that he will be remembered.”


case, integral operators the kernels of which allow a representation asRiemann-Stieltjes integrals. Stone was completely aware of the fact thatthe link between the spectral function and the solutions of (2.4) was stillmissing [47, Page 530], and this link Kodaira provided by re-consideringWeyl’s proof of the expansion theorem in the light of Stone’s analysis[32]. (Some care is needed when simplifying the expansion theorem inthe case when two singular endpoints are present but the spectrum isnevertheless simple [28, Page 206].) A particularly lucid proof of theexpansion theorem in Kodaira’s spirit was recently given in Weidmann’sbook [53, Chapter 14.1]. A recent proof which is more in the spiritof Titchmarsh can be found in [4], in which the expansion theorem isproved for any interval I, any endpoint classifications and any separatedboundary conditions, except LPC at both endpoints.

In 1987 Daphne Gilbert and D.B. Pearson identified the set of λ ∈ Rwhere Im(m(λ)) has a finite positive value as the minimal support ofthe absolutely continuous part of the measure generated by the spectralfunction ρ [19]. Using this, they proved the remarkable result that thespectrum of every self-adjoint realisation of (2.1) is absolutely continuousin an interval I if (2.4) has no subordinate solution for all λ ∈ I. A non-trivial solution u of (2.4) is called subordinate if there is a solution v of(2.4) such that

lima→∞

(∫ a

0|u|2

)(∫ a

0|v|2

)−1

= 0.

As a consequence, the absolutely continuous spectrum can be deter-mined by finding those values of λ ∈ R for which (2.4) has two linearlyindependent solutions of “comparable size”. A simplified version of theGilbert-Pearson theory can be found in Weidmann’s book [53, Chapter14.5].

9 Aftermath

The years after 1950 saw a veritable surge of papers on specific Sturm-Liouville operators and even more specific problems concerning them(the bibliography in Zettl’s book [64], far from claiming any complete-ness, lists 648 items), making Weyl’s memoir [56] one of the most fre-quently cited, but, one may venture to guess, not one of the most fre-quently read papers in spectral analysis.


10 Acknowledgements

The second author thanks H.O. Cordes, Berkeley, for a number of dis-cussions concerning the work of Hilb.

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[15] K. Friedrichs. Spektraltheorie halbbeschränkter Operatoren und Anwen-dung auf die Spektralzerlegung von Differentialoperatoren: I, II. Math.Ann. 109 (1933/34), 465-487, 685-713. (Berichtigung: Math. Ann. 110(1934/35), 777-779.)

[16] K. Friedrichs. Über die ausgezeichnete Randbedingung in der Spek-traltheorie der halbbeschränkten gewöhnlichen Differentialoperatorenzweiter Ordnung. Math. Ann. 112 (1935/36), 1–23.

[17] E. Fues. Zur Intensität der Bandenlinien und des Affinitätsspektrumszweiatomiger Moleküle. Ann. Physik (4) 81 (1926), 281-313.


[18] C. Fulton. Parametrizations of Titchmarsh’s m(λ)-functions in the limitcircle case. (Dissertation; RWTH Aachen, Germany:1973) Trans. Amer.Math. Soc. 229 (1977), 51–63.

[19] D.J. Gilbert and D.B. Pearson. On subordinacy and analysis of the spec-trum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128(1987), 30-56.

[20] E. Hellinger. Zur Stieltjesschen Kettenbruchtheorie. Math. Ann. 86(1922), 18-29.

[21] G. Hellwig. Differential operators of mathematical physics; An introduc-tion. (Translated from the German by Birgitta Hellwig. Addison-WesleyPublishing Co., Reading, Mass.-London-Don Mills, Ont. 1967.)

[22] E. Hilb. Über Integraldarstellungen willkürlicher Funktionen. Math. Ann.66 (1909), 1-66.

[23] E. Hilb. Über gewöhnliche Differentialgleichungen mit Singularitäten unddie dazugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann.76 (1915), 333-339.

[24] E. Hilb and O. Szász. Allgemeine Reihenentwicklungen. Enc. Math. Wiss.IIC11, 1229-1276, 1922. (Teubner, Leipzig: 1923-27.)

[25] D. Hilbert. Grundzüge einer allgemeinen Theorie der linearen Integral-gleichungen. (Teubner, Leipzig: 1912.)

[26] D. Hilbert und E. Schmidt. Integralgleichungen und Gleichungen mitunendlich vielen Unbekannten. Herausgegeben und mit einem Nachwortversehen von A. Pietsch. (Teubner Archiv zur Mathematik: 11, Leipzig:1989.)

[27] K. Jörgens und F. Rellich. Eigenwerttheorie gewönlicher Differentialgle-ichungen.Bearbeitet von J. Weidmann. (Springer, Berlin:1976.)

[28] I.S. Kac. The existence of spectral functions of generalized second-orderdifferential systems with a boundary condition at the singular end. Amer.Math. Soc. Transl. (2) 62 (1967), 204-262.

[29] H. Kalf. Ernst Mohrs Version der Weylschen Theorie der Sturm-Liouville-Operatoren. Sitz.-Ber. Berliner Math. Ges. 221-234. (Jahrgänge 1988-1992.)

[30] E.C. Kemble. The fundamental principles of quantum mechanics.(McGraw-Hill, New York: 1937. Dover reprint: 1958.)

[31] A. Kneser. Die Integralgleichungen und ihre Anwendungen in der math-ematischen Physik. (Vieweg, Braunschweig: 1922. 2 Aufl.)

[32] K. Kodaira. The eigenvalue problem for differential equations of the sec-ond order and Heisenberg’s theory of S-matrices. Amer. J. Math. 71(1949), 921-945.

[33] V.A. Marchenko. Sturm-Liouville operators and applications. Operatortheory: Advances and applications 22. (Birkhäuser, Basel:1986.)

[34] E. Mohr. Eine Bemerkung zur Weylschen Theorie vom Grenzkreis- undGrenzpunktfall. Ann. Mat. Pura. Appl. (4) 129 (1981), 161-199.

[35] E. Mohr. Ein Beitrag zur Weylschen Theorie vom Grenzpunktfall. Ann.Mat. Pura. Appl. (4) 132 (1982), 331-352.

[36] R. Nevanlinna. Eindeutige analytische Funktionen. Grundlehren derMath. Wiss. 46 (Springer, Berlin: 1936. 2. Aufl. 1953.)

[37] H.-D. Niessen and A. Zettl. Singular Sturm-Liouville problems: theFriedrichs extension and comparison of eigenvalues. Proc. London Math.Soc. (3) 64 (1992), 545-578.


[38] D. Race. Limit-point and limit-circle: 1910-1970. (M.Sc. thesis, Univer-sity of Dundee, Scotland, UK: 1976.)

[39] C. Reid. The life of Kurt Otto Friedrichs in Kurt Otto Friedrichs, SelectaI, 11-22. (C.S. Morawetz, Editor. Birkhäuser, Boston: 1986.) Note: IIcontains references [15] and [16] given above.

[40] F. Rellich. Die zulässigen Randbedingungen bei den singulären Eigen-wertproblemen der mathematischen Physik. (Gewöhnliche Differential-gleichungen zweiter Ordnung.) Math. Z. 49 (1943/44), 702-723.

[41] F. Rellich. Spectral theory of second-order ordinary differential equations.(Lectures delivered 1950-1951; New York University: 1953.)

[42] F. Rellich. Halbbeschränkte gewöhnliche Differentialoperatoren zweiterOrdnung. Math. Ann. 122 (1950/51), 343 -368.

[43] F. Rellich. Eigenwerttheorie partieller Differentialgleichungen. (Vorlesunggehalten im Wintersemester 1952/53 an der Universität Göttingen.)

[44] R. Rosenberger. A new characterization of the Friedrichs extension ofsemibounded Sturm-Liouville operators. J. London Math. Soc. (2) 31(1985), 501-510.

[45] A. Schneider. Eine Bemerkung zum Weyl-Stoneschen Eigenwertproblem.Arch. Math. (Basel) 17 (1966), 352-358.

[46] E. Schrödinger. Quantisierung als Eigenwertproblem. (Vierte Mitteilung)Ann. Physik (4) 81 (1926), 109-139.

[47] M.S. Stone. Linear transformations in Hilbert space and their applicationsto analysis. Amer. Math. Soc. Colloquium Pub. XV. (New York: 1932.)

[48] E.C. Titchmarsh. On expansions in eigenfunctions (IV). Quart. J. Math.Oxford. 12 (1941), 33-50.

[49] E.C. Titchmarsh. On expansions in eigenfunctions (V). Quart. J. Math.Oxford. 12 (1941), 89-107.

[50] E.C. Titchmarsh. On expansions in eigenfunctions (VI). Quart. J. Math.Oxford. 12 (1941), 154-166.

[51] E.C. Titchmarsh. Eigenfunction expansions associated with second-orderdifferential equations. I (Oxford University Press: 2nd edition:1962; 1st

edition: 1946.)[52] H-J. Vollrath. Emil Hilb (1882-1929). In P. Baumgart (Hrsg.),

Lebensbilder bedeutender Würzburger Professoren: 320-338. Degener,Neustadt/Aisch 1995.

[53] J. Weidmann. Lineare Operatoren in Hilberträumen. Teil I: Grundlagen.Teil II: Anwendungen. (Teubner, Stuttgart: 2000: 2003.)

[54] H. Weyl. Über beschränkte quadratische Formen, deren Differenz voll-stetig ist. Rend. Circ. Mat. Palermo 27 (1909), 373-392.

[55] H. Weyl. Über gewöhnliche lineare Differentialgleichungen mit singulärenStellen und ihre Eigenfunktionen. Nachr. Kgl. Ges. Wiss. GöttingenMath-Phys. Kl. (1909), 37-63.

[56] H. Weyl. Über gewöhnliche Differentialgleichungen mit Singularitätenund die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann.68 (1910), 220-269.

[57] H. Weyl. Über gewöhnliche lineare Differentialgleichungen mit singulärenStellen und ihre Eigenfunktionen. (2. Note) Nachr. Kgl. Ges Wiss. Göt-tingen Math-Phys. Kl. (1910), 442-467.

[58] H. Weyl. Zwei Bemerkungen über des Fouriersche Integraltheorem.Jahresber. Deutsch. Math.-Verein. 20 (1911), 129-141. (Berichtigung ibid.20 (1911), 339.)


[59] H. Weyl. Über das Pick-Nevanlinnasche Interpolationsproblem und seininfinitesimales Analogon. Ann. of Math. (2) 36 (1935), 230-254.

[60] H. Weyl. David Hilbert and his mathematical work. Bull. Amer. Math.Soc. 50 (1944), 612-654.

[61] H. Weyl. Ramifications, old and new, of the eigenvalue problem. Bull.Amer. Math. Soc. 56 (1950), 115-139.

[62] H. Weyl. Address of the President of the Fields medal committee 1954. In:Proc. Inter. Congress of Mathematicians: I, (1954), 161-174. (Noordhoff,Groningen, North-Holland, Amsterdam: 1957.)

[63] A. Wintner. On the location of continuous spectra. Amer. J. Math. 70(1948), 22-30.

[64] A. Zettl. Sturm-Liouville theory. Mathematical surveys and monographs,121. (Amer. Math. Soc.: 2005)

11 Remarks on the references

Weyl’s papers [54] to [58] can be found in Bd. 1, the paper [59] in Bd.3, and papers [60] to [62] in Bd. 4 of his “Gesammelte Abhandlungen”;(Springer, Berlin: 1968). The papers [56] and [59] were also includedin “Selecta Hermann Weyl”; (Birkhäuser, Basel: 1956) which were pub-lished on the occasion of his 70th birthday. Weyl added a short remarkto Satz 5 of [56], and it is with this addition that [56] was included inhis collected papers.

4From Weyl quantization to modern algebraic

index theoryMarkus J. Pflaum

Fachbereich MathematikGoethe-Universität Frankfurt/Main

[email protected]

1 Introduction

One of the most influencial contributions of Hermann Weyl to mathe-matical physics has been his paper Gruppentheorie und Quantenmechanik[We27] from 1927 and its extended version, the book [We28] which waspublished a year later and carries the same title. The main topic of thispart of Hermann Weyl’s work is the mathematics of quantum mechan-ics. After the fundamental papers by Heisenberg and Schrödinger

on the foundations of quantum mechanics had appeared in the twentiesof the last century this was the central question studied in mathemat-ical physics at that time and which to a certain degree still is presentin all attempts to construct mathematically rigorous theories unifyingquantum mechanics and general realtivity.

In his article Gruppentheorie und Quantenmechanik, Hermann Weyl

essentially introduced two novel aspects to the mathematics of quantummechanics, namely the following:

(i) The representation theory of (compact) Lie groups on Hilbertspaces was applied to mathematically determine atomic spectra.

(ii) A conceptually clear quantization method was proposed whichassociates quantum mechanical operators to classical observableswhich mathematically are represented by appropriate functions ofthe space and momentum variables. Nowadays, this quantizationscheme is named after his inventor Weyl quantization.

In this paper I will elaborate only on the second aspect, since therepresentation theory of compact Lie groups has already been coveredin detail in other contributions to these proceedings.

Interestingwise, other than the group theoretical part in Weyl’s ar-

84

From Weyl quantization to modern algebraic index theory 85

ticle from 1927, his quantization method did not immediately find ac-ceptance in the scientific community as the following part from a reviewby John von Neumann in Zentralblatt shows:

Sodann wird eine Zuordnungsvorschrift von Matrizen zu beliebigen klassischenGrößen (d.h. Funktionen der Koordinaten und Impulse) vorgeschlagen. (Dasie indessen gewisse wesentliche Anforderungen, die an eine solche Zuordnungzu stellen sind – z.B. die Definität der Matrix für wesentlich nichtnegativeGrößen u.ä. – verletzt, hat sie sich, trotz ihres einfachen und eleganten Baues,nicht durchsetzen können.)

Only much later after the invention of pseudodifferential operators[Hö] and deformation quantization [BFFLS] the virtue and power ofWeyl quantization became fully clear. As we will see in Section 4 ofthis article one can namely show by using the modern language of pseu-dodifferential operators that Weyl quantization satisfies the axioms ofa deformation quantization á la [BFFLS] (cf. [Pf98, NeTs96]). Myimpression is that H. Weyl with his vision for a mathematically soundquantization scheme was quite ahead of his time. The following quotefrom the book [We28] supports this impression:

Ich kann es nun einmal nicht lassen, in diesem Drama von Mathematik undPhysik – die sich im Dunkeln befruchten, aber von Angesicht zu Angesicht sogerne einander verkennen und verleugnen – die Rolle des (wie ich genügsamerfuhr, oft unerwünschten) Boten zu spielen.

Generalizing Heisenberg’s commutation relations, P. M. Dirac pro-posed in his influential book [Di, §. 21] that a quantization map q whichassociates to every classical observable a an element q(a) of an algebraof quantum mechanical observables should satisfy the following commu-tation relation: [

q(a), q(b)]

= iq(a, b), (1.1)

where denotes Planck’s constant divided by 2π, a, b are classical ob-servables, and −,− is the Poisson bracket. As Dirac noticed, thecommutation relations (1.1) show that “classical mechanics may be re-garded as the limiting case of quantum mechanics when tends to zero”(cf. [Di, §. 21]). For physical reasons Dirac’s quantization conditionsare usually supplemented by the requirement that the algebra of quan-tum observables q(a) acts irreducibly on a Hilbert space H. This Hilbertspace H or more precisely the corresponding projective space PH of raysin H is then interpreted as the space of (pure) states of the quantummechanical system.

86 Markus J. Pflaum

Let me explain now from the point of view of a mathematician what one meansby quantization. This can be seen most easily by the following diagram:

classical mechanics quantum mechanics

states

points x of a sym-plectic manifold Mrespectively propa-bility distributionson M

−→positive normed li-near functionals µon a noncommuta-tive C∗-algebra A

observableselements a of thePoisson algebra(C∞(M ), , )

←−(self-adjoint) el-ements a of theC∗-algebra A

measuringprocess

evaluation(x, a) → a(x)

evaluation(µ, a) → µ(a)

The arrow from left to right is given by quantization while the arrow in theopposite direction is given by a classical limit process.

In 1946 it has been observed by Groenewold [Gr] and later refinedby van Hove [Ho] that for the algebra of (polynomial) observableson R2n with its standard Poisson bracket a quantization map fulfillingDirac’s commutation relations Eq. (1.1) together with the irreducibil-ity condition cannot exist. The theorems by Groenewold–van Hove

were extended by Gotay et al. [GoGrHu, Go] to more general sym-plectic manifolds. By all these no go results the question arises, whatconditions a reasonable quantization theory should satisfy then.

Weyl’s quantization scheme motivated the right answer to that prob-lem. As it has been pointed out by Bayen, Flato, Fronsdal, Lich-

nerowicz and Sternheimer in [BFFLS], one should regard quanti-zation as a formal deformation of the algebra of classical observableson a symplectic manifold in the sense of Gerstenhaber [Ge]. Thismeans that Dirac’s quantization condition is required to hold only up tohigher order in . Weyl quantization satisfies this requirement and thusprovides an important example of a deformation quantization.

The paper [BFFLS] initiated quite an amount of research on the exis-tence and uniqueness of deformation quantizations. The most outstand-ing are probably the existence theorem for deformation quantizations


over a symplectic manifold by deWilde–Lecomte [deWiLe], the geo-metric and intuitive construction of star products in the symplectic caseby Fedosov [Fe94], and the result on the existence and the classifica-tion of deformation quantizations for Poisson manifolds by Kontsevich

[Ko]. For a detailed overview on this see for example [DiSt].

2 Weyl’s commutation relations

In his analysis of quantization Weyl started from the Heisenberg com-mutation relations [

P,Q]

= −i, (2.1)

where Q resp. P denotes the quantum mechanical space resp. momen-tum operator. Weyl showed that these relations cannot be realized bybounded operators on a Hilbert space. His idea was then to integratethe Heisenberg commutation relations which leads to the relations

V (s)U(t) = e−istU(t)V (s), s, t ∈ R, (2.2)

where V (s) = eisQ is the unitary abelian group generated by Q, andU(t) = eitP the one generated by P . For the Schrödinger representationon L2(R) given by

Qu(x) = xu(x), P u(x) = −idu

dx(x) for u ∈ S(R) and x ∈ R

(2.3)

one knows that(V (s)u

)(x) = eisxu(x) and

(U(t)u

)(x) = u(x + t). (2.4)

One thus obtains the integrated Schrödinger representation which ob-viously satisfies the Weyl commutation relations (2.2). Up to unitaryequivalence, the integrated Schrödinger representation is the only ir-reducible nontrivial representation of the Weyl commutation relations.Note that the Heisenberg commutation relations have more than justone equivalence class of irreducible representations by (necessarily un-bounded) symmetric operators on a Hilbert space (see [Schm]).

Using the integrated Schrödinger representation let us define now thefollowing projective representation of R2 :

W (s, t) = e−i2 stU(t)V (s). (2.5)

88 Markus J. Pflaum

For a ∈ S(R2), the space of Schwarz test functions on R2 , define itsWeyl quantization qW(a) : C∞cpt(R) → C∞(R) by⟨

v, qW(a)u⟩

:=∫

R2a(s, t)〈v,W (s, t)u〉 ds dt, u, v ∈ C∞cpt(R), (2.6)

where a denotes the Fourier transform of a. This is the original formof Weyl quantization. Let us rewrite it in a more convenient form byapplying the transformation rule and Fourier transformation:

〈v, qW(a)u〉 =∫

R2a

∫R

v(x)(W (s, t)u

)(x) dx ds dt

=∫

R3a(s, t)v(x) e−

i2 st

(U(t)V (s)u

)(x) dx ds dt

=∫

R3a(s, t)v(x) e−

i2 st eis(x+t) u(x + t) dx ds dt

=1

∫R3

a(s,t

)v(x)eis(x+t/2) u(x + t) dx ds dt

=1

2π

∫R3

a(x +

t

2, ξ)v(x)e−

i

tξ u(x + t) dx dξ dt

=1

2π

∫R3

v(x)e−i

tξ a(x

2+(x + t

2, ξ))

u(x + t) dx dξ dt

=1

2π

∫R3

v(x)ei

(x−y )ξ a(x + y

2, ξ)u(y) dx dy dξ

=1

2π

⟨v,

∫R2

ei

(•−y )ξ a(•+ y

2, ξ)u(y) dy dξ

⟩,

hence [qW(a)u

](x) =

12π

∫R2

ei

(x−y )ξ a(x + y

2, ξ)u(y) dy dξ, (2.7)

which is the form of the Weyl quantization as it usually can be foundin the literature. As one checks immediatley, qW(a) is a densly defined(in general unbounded) linear operator on L2(R) which is symmetric, incase a is a real-valued function. If one interprets the right hand side ofEq. (2.7) as an oscillatory integral (see [GrSj]), then Eq. (2.7) defineseven for symbols a ∈ S∞(R) (cf. Sec. 4) a quantized observable qW(a)which by definition then is a pseudodifferential operator on R. By astandard argument in pseudodifferential calculus one shows that

qW(a) qW(a)− qW(b) qW(b) = −iqW(a, b) + o(2) for a, b ∈ S∞,

which means that Weyl quantization satisfies Dirac’s quantization con-dition up to higher orders in or in other words that the algebra of


pseudodifferential operators is a deformation of the algebra of symbolsin direction of the Poisson bracket. Let us now explain the concept of adeformation quantization in some more detail.

3 Deformation quantization

Definition 1 ([BFFLS]) By a deformation quantization of a symplec-tic manifold (M,ω) one understands an associative and C[[]]-bilinearproduct on the space A := C∞(M)[[]] of formal power series in the(now formal) variable and with coefficients in the space C∞(M) suchthat the following axioms hold true:

(i) There exist bidifferential operators ck on M such that a b =∑∞k=0 ck (a, b) k for all a, b ∈ C∞(M) and such that c0 is the

commutative pointwise product of smooth functions on M .(ii) One has a 1 = 1 a = a for all a ∈ C∞(M).

(iii) The commutation relation

[a, b] = −ia, b+ o(2)

is satisfied for all a, b ∈ C∞(M), where [a, b] := a b− b a.

The product is also called a star-product on M .

Example 3.1 Consider a finite dimensional symplectic vector space(V, ω), and let

−,− : C∞(V )⊗ C∞(V ) → C∞(V ), a⊗ b →∑

1≤i,j≤dim V

Πij∂a

∂xi

∂b

∂xj

be its Poisson bracket, where (xi)1≤i≤dim V denote some coordinates ofV . Since the standard Poisson bivector Π :=

∑Πij

∂∂xi⊗ ∂

∂xjis constant,

the operator

Π : C∞(V )⊗C∞(V ) → C∞(V )⊗C∞(V ), a⊗b →∑

1≤i,j≤2n

Πij∂a

∂xi⊗ ∂b

∂xj

is well-defined. Denoting by µ the pointwise product of functions, onecan now put

: C∞(V )[[]]⊗ C∞(V )[[]] → C∞(V )[[]],

a⊗ b →∑k∈N

(−i)k

k!µ(Πk (a⊗ b)

)

90 Markus J. Pflaum

and thus obtains a star product on C∞(V ), the so-called Weyl–Moyal-product. It is immediately checked that (C∞(V )[[]], ) is a deformationqunatization in the above sense.

By construction, the Weyl–Moyal-product makes sense also on thespace WV of formal power series in with formal power series at theorigin of V as coefficients. One calls the resulting algebra (WV, ) theformal Weyl algebra of V , and obtains an epimorphism of algebras

(C∞(V )[[]], ) → (WV, )

given by formal power series expansion at the origin in each degree of .

The existence of a star product on an arbitrary symplectic manifoldwas an open mathematical problem for almost ten years after the article[BFFLS] had appeared, and was settled by independant methods inthe papers [deWiLe] and [Fe94]. Another ten years passed until theexistence of star products on Poisson could be proved in [Ko].

Let us briefly sketch the main idea of the proof by Fedosov [Fe96],since his approach contains essential tools which lead to algebraic indextheory. Consider a symplectic manifold (M,ω) and its tangent bun-dle TM . Since each of the tangent spaces TpM , p ∈ M is a sym-plectic vector space, one can form the bundle of formal Weyl alge-bras WM :=

⊔p∈M WTpM . Note that the bundle of formal Weyl

agebras is well-defined because the symplectic group acts as automor-phisms on the formal Weyl algebra. Now consider the bundle of formsΛ•WM := WM⊗Λ•M . Its space of smooth sections Ω•W(M) obviouslyis a noncommutative algebra with product denoted by •. The funda-mental observation by Fedosov was that for an appropriate flat gradedderivation D with respect to the product • on Ω•W(M) the subalgebra

WD (M) := s ∈ Ω0W(M) | Ds = 0

of flat sections is linearly isomorphic to the space of formal power seriesC∞(M)[[]]. Via the resulting isomorphism q : C∞(M)[[]] → WD (M)one can then push down the product on WD (M) to C∞(M)[[]] andthus obtains a star product on M . The flat connection needed for thisconstruction has the form

D = ∇+ [A,−],

where ∇ is a symplectic connection on M , i.e. ∇ω = 0, [−,−] is thecommutator with respect to the product •, and A ∈ Ω1W(M). The


cohomology class of the curvature

Ω := ∇A +12[A,A]

(note that it is a formal power series in ) classifies the star product

up to equivalence.For the application of deformation quantization to index theory, the

notion of a trace on a deformation quantization(C∞cpt(M)[[]],

)is cru-

cial. By a that one understands a linear functional tr : C∞cpt(M)[[]] →C[[, −1 ] which vanishes on commutators. The following result providesessential information on the existence and uniqueness of such traces.

Proposition 3.1 ([NeTs95, Fe96]) The space of traces on a deforma-tion quantization

(C∞cpt(M)[[]],

)over a connected symplectic manifold

M is one-dimensional.

4 Pseudodifferential operators

Next we will set up Weyl quantization within the language of pseudo-differential operators. Before we come to the details of this let us recallsome basics of that theory.

Let U ⊂ Rn be open. By a symbol on U × RN of order m ∈ R oneunderstands a function a ∈ C∞(U × RN ) such that for every compactK ⊂ U and all multi-indices α ∈ Nn and β ∈ NN there exists a CK,α,β >

0 such that∣∣∣∂αx ∂β

ξ a(x, ξ)∣∣∣ ≤ CK,α,β (1 + ||ξ||)m−|β | for all (x, ξ) ∈ K × RN . (4.1)

The space of such symbols is denoted by Sm (U, RN ). Obviously, one caneasily generalize the notion of a symbol of order m to smooth maps a :E → R defined on a vector bundle E →M by requiring Eq. (4.1) to holdlocally in bundle charts. For every manifold X we denote the space ofsymbols of order m on the cotangent bundle T ∗X by Sm (X). Moreover,one puts S∞(X) :=

⋃m∈R Sm (X) and S−∞(X) :=

⋂m∈R Sm (X).

A pseudodifferential operator over U ⊂ Rn now is a linear operatorA : C∞cpt(U) → C∞(U) which can be represented as an oscillatory integral(cf. [GrSj, Sec. 1])

Au(x) =1

(2π)n

∫Rn

∫U

ei〈x−y ,ξ〉a(x, y, ξ)u(y) dy dξ, (4.2)

where u ∈ C∞cpt(U) and a ∈ Sm (U × U, Rn ) for some m ∈ R ∪ ±∞.

92 Markus J. Pflaum

The space of thus defined pseudodifferential operators of order m willbe denoted by Ψm (U). More generally, if X is a manifold, the spaceΨm (X) of pseudodifferential operators of order m on X consists of alllinear operators A : C∞cpt(X) → C∞(X) which can be written in the form

Au = A0u +∑j∈J

ϕj

(Aj

((ϕju) x−1

j

)) xj ,

where the xj , j ∈ J run through an atlas of X, (ϕj )j∈J is a locally fi-nite smooth partition of unity subordinate to the domains of the chartsϕj , the Aj are pseudodifferential operators on Rdim X of order m, andfinally A0 is a smoothing operator, which means that its Schwartz kernelis smooth. Let us restrict our considerations now to the space Ψ∞

ps (X) ofproperly supported pseudodifferential which means of all pseudodiffer-ential operators A such that the projections pr1/2 : suppKA → X of thesupport of the Schwartz kernel of A on the first resp. second coordinateare proper maps. Since every properly supported pseudodifferential op-erator maps functions with compact support to functions with compactsupport, Ψ∞

ps (X) turns out to be an algebra which, as we will see in thefollowing, can be interpreted as a quantization of the symbol algebra onTX.

Let us provide some details. After the choice of a riemannian metricon X and fixing an ordering parameter s ∈ [0, 1], define for every symbola ∈ Sm (X) and ∈ R∗ a quantization qs(a) : C∞cpt(X) → C∞cpt(X) by

[qs(a)u](x) =1

(2π)n

∫T ∗X

χ(x, y) ei〈exp−1

y (x),ξ〉a(τgs (x,y ),y ξ

)u(y) dydξ.

(4.3)The ingredients of this formula are given as follows. As usual, exp de-notes the exponential function with respect to the riemannian metricon X, and χ is a properly supported cut-off function around the diago-nal of X ×X such that exp−1

y (x) is defined for all (x, y) ∈ suppχ. Bygs(x, y) we mean the s-midpoint between x and y, or in other words thepoint exp

(s · exp−1

x (y)). For x and y close enough we denote by τx,y the

parallel transport in T ∗X from T ∗y X to T ∗

x X along the geodesic joiningx and y. Finally, dy dξ stands for the Liouville volume element on thesymplectic manifold T ∗X. One checks immediately (cf. [Pf98, Vo]) thatqs(a) is a (properly supported) pseudodifferential operator of order m.In case s = 0 one calls it the standard order quantization of the symbola, if s = 1

2 , one obtains Weyl quantization on the riemannian manifoldX. The reader is invited to check that on the cotangent bundle of R,


q 12

coincides with the Weyl quantization qW from Eq. (2.7) (up to somenegligible smoothing operator).

The quantization map qs has a pseudoinverse, namely the symbol mapσs : Ψ∞

ps (X) → S∞(X) which is defined by

σs(A)(x, ξ) =∫Tx X

χs(x, v) ei〈v ,ξ〉 KA

(expx(−sv), expx((1− s)v)

)ρs(x, v)dθx(v),

where KA is the Schwartz kernel of the operator A, the cut-off functionχs is defined by χs(x, v) := χ

(expx(−sv), expx((1 − s)v)

), θx is the

euclidean volume element of TxX induced by the riemannian metric onX, and the metric factor ρs satisfies ρs(x, v) = ρ

(expx(−sv), expx((1−

s)v))

with ρ(x, expx v)θx =(exp∗

x µ)(v), and µ the riemannian volume

element on X. Then one has

σs qs(a)− a ∈ S−∞(X) and qs σs(A)−A ∈ Ψ−∞ps (X) (4.4)

for all symbols a and pseudodifferential operators A on X, which showsthat they are inverse to each other up to smoothing operators resp. sym-bols. Moreover, one can prove (cf. [Pf98]) that

[qs(a), qs(b)] = −i qs(a, b) + o(2) (4.5)

for all symbols a, b. This means that each qs and in particular Weylquantization qW := q 1

2induces a deformation quantization of the cotan-

gent bundle TX.Another usefull feature of the quantization qs is that it allows to com-

pute the operator trace of qs(a) for every symbol a of order m < dim X.According to [Pf98, Vo] this trace is given by

trqs(a) =1

(2π)dim X

∫T ∗X

aωdim X , (4.6)

where ω denotes the canonical symplectic form on the cotangent bundleT ∗X.

5 The algebraic index theorem

Let us first recall some basic notions from index theory of Fredholm op-erators. Assume to be given two Hilbert spaces H1 , H2 and a Fredholmoperator F : H1 → H2 , which means that F is a linear operator whichhas finite dimensional kernel and cokernel. Its index is then defined as

94 Markus J. Pflaum

the integer

indF := dim ker F − dim coker F. (5.1)

The index has the following crucial properties:

• it is homotopy invariant, i.e.

indF (0) = indF (1)

for every continuous path F : [0, 1] → Fred(H1 ,H2) of Fredholmoperators,

• it is additive with respect to composition, i.e.

ind(F1 F2) = indF1 + indF2

for two composable Fredholm operators F1 and F2 , and finally• the index is invariant under compact perturbations, i.e.

ind(F + K) = indF

for every Fredholm operator and every compact operator K fromH1 to H2 .

Every Fredholm operator F : H1 → H2 has a pseudoinverse R : H2 →H1 , which in other words means that idH1 − R F and idH2 − R F

are both compact operators. One can even choose R such that theseoperators are of trace class. Then one can compute the index of F bythe following formula (cf. [Fe96]):

indF = tr(idH1 −R F )− tr(idH2 −R F ). (5.2)

In the theory of linear partial differential equations, Fredholm opera-tors appear abundantly. Namely, if E → X is a (metric) vector bundleover a compact (riemannian) manifold X, and D : Γ∞(E) → Γ∞(E)an elliptic differential operator (which means that its principle symbolis invertible) then it induces a Fredholm operator between appropriateSobolev completions of Γ∞(E). In particular, D then has a finite index,and this index does not depend on the particular choice of a Sobolev com-pletion. By the celebrated index theorem of Atiyah–Singer [AtSi], theindex of D can be computed by topological data as follows:

indD = (−1)dim X

∫X

Ch(σp(D)) td(TCX), (5.3)

where σp(D) denotes the principal symbol of the differential operator D,Ch its Chern character and td(TCX) is the Todd class of the complexifiedtangent bundle.


As it has been observed by Fedosov [Fe96] and Nest–Tsygan

[NeTs95], an “algebraic” version of this index theorem can be formulatedand proved within the framework of deformation quantization. Recallthat the the index of an elliptic operator can be computed by Eq. (5.2).If one interprets a deformation quantization as a kind of “formal pseu-dodifferential calculus”, it makes sense to consider elliptic elements inthe deformed algebra and define an algebraic index for these objects.More precisely, given a symplectic manifold M with a star product ,one understands by an elliptic pair in C∞(M)[[]] a pair of projectionsP,Q in the matrix algebra over (C∞(M)[[]], ) such that the differenceP −Q has compact support. This means in particular, that every ellip-tic pair (P,Q) determines an element [P ] − [Q] of the K-theory of thedeformed algebra. The algebraic index of the K-theory class [P ] − [Q]of an elliptic pair is defined by

inda

([P ]− [Q]

):= tr(P −Q), (5.4)

where tr is the (up to normalization) unique trace on the matrix algebraover C∞(M)[[]] (cf. Prop. 3.1). Note that the index is indeed well-defined on the K-theory of C∞(M)[[]].

The space of equivalence classes [P ]− [Q] of elliptic pairs is isomorphicto the space of equivalence classes of elliptic quadruples. These objectswere introduced by Fedosov [Fe96] and are the natural generalizationsof elliptic operators to star product algebras. More precisely, an ellipticquadruple is a quadruple (D,F, P , Q) of elements of the matrix algebraover C∞(M)[[]] such that the following holds:

(i) P and Q are projections.(ii) The elements P−DR and Q−RD have both compact support.

The element D of an elliptic quadruple hereby generalizes an ellipticpseudodifferential operator on a closed manifold, and F can be inter-preted as its quasi-inverse.

By Eq. (5.2), the following definition of the algebraic index of an(equivalence class of an) elliptic quadruple appears to be reasonable:

inda

([D,F, P , Q]

):= tr(Q−R D)− tr(P −D R). (5.5)

In fact, one can show that the thus defined algebraic index is com-patible with the algebraic index on elliptic pairs under the mentionedisomorphism between equivalence classes of elliptic pairs and of ellipticquadruples.

96 Markus J. Pflaum

There is another remark in order, here. For an elliptic pair (P,Q)the coefficients (P0 , Q0) of order 0 in the expansion in powers of areobviously projections in the matrix algebra over C∞(M), and the virtualbundle [P0 ] − [Q0 ] defines a K-theory class of M . This map turns outto be an isomorphism between K-theories, which shows that K-theoryis invariant under deformation (cf. [Ro], [Fe96, Sec. 6.1]).

The main result of algebraic index theory is the theorem below. Byapplication of this algebraic index theorem to cotangent bundles of com-pact riemannian manifolds and the deformation quantization inducedby Weyl quantization one obtains another proof of the index formula byAtiyah–Singer.

Theorem 5.1 ([NeTs95, Fe96]) Let M be a symplectic manifold witha deformation quantization . The algebraic index of an elliptic pair[P ]− [Q] is then given by

inda

([P ]− [Q]

)=∫

M

Ch([P0 ]− [Q0 ]

)exp

(− Ω

2π

)A(M), (5.6)

where A(M) denotes the A-genus of M , and Ω the characteristic classof the star product on M

The proof of the algebraic index theorem is quite involving. In theapproach by Nest–Tsygan, methods from cyclic homology theory andLie algebra cohomology are used intensively. We refer to the originalliterature for that.

6 The algebraic index theorem for orbifolds

The problem to generalize index theorems to spaces more general than(compact) manifolds has been an active area of mathematical researchsince many years. For orbifolds, a class of singular spaces which hasattained much interest in geometry and mathematical physics, an alge-braic index theorem can be proved.

In local charts, orbifolds are represented as quotients of manifolds byfinite groups. Globally, and that is the approach we use in our setup,orbifolds can be presented as orbits of proper étale Lie groupoids G (see[Mo] for details). The concepts of a deformation quantization, of vectorbundles, and of K-theory can all be generalized to orbifolds by requiringthe objects (like a star product or a vector bundle) to be invariant on therepresenting proper étale groupoid. For example, the orbifold K-theory


K0orb(X) of an orbifold X consists of equivalence classes of equivariant

virtual bundles on the representing groupoid.A quite usefull object associated to an orbifold X is its inertia orbifold

X. Locally, X consists of all fixed point manifolds of the locally repre-senting orbifold charts. The inertia orbifold carries a lot of informationabout the singularities of the orbifold. The connected components of theinertia orbifold X are sometimes called the sectors of the orbifold X.

The orbifold case is different to the manifold case in particular by oneimportant aspect. The dimension of traces on a deformation quantiza-tion on a symplectic orbifold is in general not one (even if X is con-nected), but given by the number of sectors [NePfPoTa]. This meansthat one has to single out a particular trace to define the algebraic indexfor a deformation quantization on an orbifold. Fortunately, there existsa kind of “universal” trace on an orbifold, which captures from each sec-tor a normalized contribution. With that universal trace the followingalgebraic index formula can be proved.

Theorem 6.1 ([PfPoTa]) Let M be a symplectic orbifold presentedby a proper étale Lie groupoid G carrying a G-invariant symplectic formω. Let be a star product on M , and let E and F be G-vector bundleswhich are isomorphic outside a compact subset of M . Then the followingformula holds for the index of [E]− [F ] ∈ K0

orb(M):

tr∗([E]− [F ]

)=∫

M

1m

Chθ

(RE

2πi −RF

2πi

)det

(1− θ−1 exp(−R⊥

2πi )) A

(R⊥

2πi

)exp

(− ι∗Ω

2πi

),

(6.1)where tr : C∞cpt[[]] G → C[[, −1 ] is the universal trace on the con-volution algebra capturing from each sector one contribution, m is a lo-cally constant combinatorial function measuring the order of the isotropygroup, and Ω is the characteristic class of the deformation quantiza-tion. The symbol θ denotes the action of the local isotropy groups, andChθ

(RE

2πi −RF

2πi

)is the equivariant Chern character which à la Chern-

Weil is determined by equivariant curvatures RE and RF . Finally R⊥

denotes the curvature of the normal bundle of the local embedding of X

into X.

Like in the manifold case one can construct a symbol calculus andWeyl quantization for orbifolds. Since Weyl quantization on an orbifoldX defines a deformation quantization over the symplectic orbifold T ∗X,one can then derive an analytic index formula from the algebraic index

98 Markus J. Pflaum

theorem for orbifolds. One then obtains the Kawasaki index formulafor orbifolds (see [Ka] and [Fa]).

Bibliography

[AtSi] Atiyah, M.F. and I.M. Singer: The Index of Elliptic Operators I.Ann. Math. 87, 484–530 (1968).

[BFFLS] Bayen, F., M. Flato, C. Fronsdal, A. Lichnerowicz, andD. Sternheimer: Deformation theory and quantization, I and II .Ann. Phys. 111 (1978), 61–151.

[Di] Dirac, P.A.M.: The Principles of Quantum Mechanics. 4th ed. Oxford,Clarendon Press, 1947.

[DiSt] Dito, G., and Sternheimer, D.: Deformation quantization: gen-esis, developments and metamorphoses. in “Deformation quantization”(Strasbourg, 2001), 9–54, IRMA Lect. Math. Theor. Phys., 1, de Gruyter,Berlin, 2002.

[Fa] Farsi, C.: K-theoretical index theorems for orbifolds,Quart. J. Math. Oxford Ser. (2) 43, no. 170, 183–200 (1992).

[Fe94] Fedosov, B.: A simple geometrical construction of deformationquantization, J. Diff. Geom. (1994).

[Fe96] Fedosov, B.: Deformation Quantization and Index Theory,Akademie Verlag, 1996.

[Ge] Gerstenhaber, M.; On the deformations of rings and algebras, Ann.of Math. 79, 59–103 (1964).

[Go] Gotay, M.J.: Obstructions to quantization. in “Mechanics: from theoryto computation. Essays in honor of Juan-Carlos Simo.” Papers invited byJournal of Nonlinear Science editors. Springer, New York. 171–216 (2000).

[GoGrHu] Gotay, M. J., H. Grundling and Hurst, C.A.: AGroenewold-Van Hove theorem for Ssp2. Trans. Am. Math. Soc. 348,no. 4, 1579–1597 (1996).

[GrSj] Grigis A., and J. Sjøstrand: Microlocal Analysis for DifferentialOperators., London Mathematical Society Lecture note series, vol. 196,Cambridge University Press, 1994.

[Gr] Groenewold, H.J.: On the principles of elementary quantum mechan-ics. Physics 12, 405–460 (1946).

[Hö] Hörmander, L.: Pseudodifferential operators Comm. PureAppl. Math. 18, 501–517 (1965).

[Ho] van Hove, L.: Sur le problème des relations entres les transformationsunitaires de la mécanique quantique et les transformation canoniques dela mécanique classique. Acad. Roy. Belgique Bull. Cl. Sci. (5) 37, 610–620(1951).

[Ka] Kawasaki, T.: The index of elliptic operators over V-manifolds,Nagoya Math. J. 84, 135-157 (1981).

[Ko] Kontsevich, M.: Deformation quantization of Poisson manifolds, I,arXiv:q-alg/9709040 (1997).

[Mo] Moerdijk, I.: Orbifolds as groupoids: an introduction, Adem, A. (ed.)et al., Orbifolds in mathematics and physics (Madison, WI, 2001), Amer.Math. Soc., Contemp. Math. 310, 205–222 (2002).


[NePfPoTa] Neumaier, N., M. Pflaum, H. Posthuma and X. Tang:

Homology of of formal deformations of proper étale Lie groupoids, Journalf. die reine und angewandte Mathematik 593 (2006).

[NeTs95] Nest, R., and B. Tsygan: Algebraic index theorem, Comm.Math. Phys 172, 223–262 (1995).

[NeTs96] Nest, R., and B. Tsygan: Formal versus analytic index theorems,Intern. Math. Research Notes 11, 557–564 (1996).

[Pf98] Pflaum, M.J.: A deformation-theoretical approach to Weyl quantiza-tion on riemannnian manifolds, Lett. Math. Physics 45 277–294 (1998).

[PfPoTa] Pflaum, M., H. Posthuma and X. Tang: An algebraic indextheorem for orbifolds. Adv. Math. 210, 83–121 (2007).

[Ro] Rosenberg, J.: Behavior of K-theory under quantization, in “OperatorAlgebras and Quantum Field Theory”, eds. S. Doplicher, R. Longo, J. E.Roberts, and L. Zsido, International Press, 404–415 (1997).

[Schm] Schmüdgen, K.: On the Heisenberg commutation relation. II. Publ.Res. Inst. Math. Sci. 19, no. 2, 601–671 (1983).

[Vo] Vornov, T.: Quantization of forms on the cotangent bundle. Comm.Math. Phys. 205, no. 2, 315–336 (1999).

[We27] Weyl, H.: Quantenmechanik und Gruppentheorie. Z. f. Physik 46,1–46 (1927).

[We28] Weyl, H.: Quantenmechanik und Gruppentheorie. S. Hirzel Verlag,Leipzig, (1928).

[deWiLe] De Wilde, M., and P. Lecomte: Formal deformations of thePoisson Lie algebra of symplectic manifold and star-products. Existence,equivalence, derivations, in “Deformation Theory of Algebras and Struc-tures and Applications” (Dordrecht) (M. Hazewinkel and M. Gersten-haber, eds.), Kluwer Acad. Pub., 1988, pp. 897–960.

5Sharp spectral inequalities for the Heisenberg

LaplacianA. M. Hansson

Department of Mathematics, [email protected]

A. LaptevDepartment of Mathematics

Imperial College, [email protected] & [email protected]

Abstract We obtain sharp inequalities for the spectrum of the Heisen-berg Laplacian with Dirichlet boundary conditions in 2N+1-dimensionaldomains of finite measure. We also give another proof of an inequalityobtained by Strichartz.

1 Introduction

The aim of this paper is to prove uniform inequalities for a class of differ-ential operators with discrete spectrum. In particular, we give a simpleproof of a previously known result by Strichartz [24] for hypoellipticoperators.

Let X1 and X2 be two vector fields in R3 , expressed in coordinatesx = (x1 , x2 , x3) as

X1 := ∂x1 +12x2∂x3 , X2 := ∂x2 −

12x1∂x3 . (1.1)

The Lie algebra of left-invariant vector fields on the first Heisenberggroup H1 is spanned by X1 , X2 and X3 := ∂x3 . H1 may be describedas the set R2 × R equipped with the group law

(x1 , x2 , x3)(y1 , y2 , y3) =(

x1 + y1 , x2 + y2 , x3 + y3 +x1y2 − x2y1

2

).

These vector fields satisfy the canonical commutation relation

[X1 ,X2 ] = −X3 .

The quadratic form

a[u] :=∫

R3

(|X1u(x)|2 + |X2u(x)|2

)dx, u ∈ H1(R3), (1.2)

100

Sharp spectral inequalities for the Heisenberg Laplacian 101

defines a self-adjoint operator

A := X∗1 X1 + X∗

2 X2 = −X21 −X2

2

from the Kolmogorov-Hörmander class [8] of second-order hypoellipticoperators. A, usually referred to as the Heisenberg Laplacian, is non-elliptic because it is associated to a degenerate metric on H1 , see [26].The operators

Z :=X1 + iX2√

2and Z :=

X∗1 − iX∗

2√2

are related to creation and annihilation operators; their correspondingenergy operator ZZ + ZZ coincides with A.

Let Ω ⊂ R3 be a domain of finite measure. The closure of the form(1.2), defined on the class of functions C∞

0 (Ω), gives us a self-adjoint op-erator AΩ which is identified with the operator A with Dirichlet bound-ary conditions on the boundary ∂Ω. It is well known that the spectrumof this operator is discrete and the corresponding eigenvalues (λk )∞k=1accumulate at ∞.

In a later section we will also study the elliptic operator

B := −X21 −X2

2 −X23 in L2(R3),

which commutes with A and hence acts on each of its eigenspaces. Bythe approach used to prove the trace inequality for A we will obtainan estimate which turns out to be identical to the classical Weyl-typeasymptotics.

2 Hypoelliptic case: main results

The aim of this article is to obtain a sharp uniform spectral inequalityof Berezin-Li-Yau type for the trace Tr ϕλ (AΩ). Let

ϕλ(t) = (λ− t)+ =

λ− t, λ > t,

0, λ ≤ t.

Theorem 2.1 Let Ω ⊂ R3 be a domain of finite measure and let AΩ

be the operator A with Dirichlet boundary conditions in Ω. Then thespectrum of AΩ is discrete and

Tr ϕλ(AΩ) =∑

k

(λ− λk )+ ≤196|Ω|λ3 . (2.1)

102 A.M. Hansson and A. Laptev

Remark 2.2 The operator A is unitarily equivalent to a two-dimensionalLaplacian with constant magnetic field, see (6.1). Note, however, that(2.1) cannot be considered as the (semi-classical) phase-volume estimate,which is infinity in the case of the Heisenberg Laplacian with Dirichletboundary conditions.

Remark 2.3 For the Dirichlet Laplacian with constant magnetic fieldthe uniform inequalities for the eigenvalues have been obtained in [6] and[7].

This result implies several corollaries.

Corollary 2.1 Let N(λ) be the counting function of AΩ ,

N(λ) := k : λk < λ. (2.2)

Then under the conditions of Theorem 2.1 we have

N(λ) ≤ 927 |Ω|λ

2 .

Remark 2.4 Uniform spectral inequalities for the number of eigenvaluesbelow λ > 0 for Dirichlet Laplacian were obtained by Pólya [21] for tilingdomains, in [4] and [5] for bounded domains and in [16], [M] and [23] fordomain of finite Lebesgue measure. The sharp constant in this inequalityremains unknown.

Applying the Aizenmann-Lieb principle [1] for γ > 1 and an argumentclose in spirit [7] for γ < 1, we obtain

Corollary 2.2 Under the conditions of Theorem 2.1 and for any γ > 0,the eigenvalues of the Dirichlet Heisenberg Laplacian satisfy the Lieb-Thirring inequality ∑

k

(λ− λk )γ+ ≤ Kγ |Ω|λγ+2

with

Kγ =

932

γγ

(γ + 2)γ+2 , 0 < γ ≤ 1,

116

1(γ + 1)(γ + 2)

, 1 ≤ γ.(2.3)

Remark 2.5 Note that limγ↓0 Kγ is the constant in Corollary 2.1. How-ever, taking this as the initial value and applying the technique we used to


estimate Kγ ‘upwards’ would have given 2−69(γ +1)−1(γ +2)−1 , γ < 1,which decreases at a slower rate.

In the case where ∂Ω is smooth, it has been obtained in [19] that thefollowing spectral asymptotic formula holds true:

limλ→∞

λ−2N(λ) =∫

Ωγ(x)dx.

Here γ is a continuous and strictly positive function, which is givenrather implicitly. Theorem 2.1 and Corollaries 2.1, 2.2 allow us to extendthis statement, with an explicit right-hand side, to arbitrary domains offinite measure with non-smooth boundaries. Approximating the domainby cylinders and arguing by the variational principle, we obtain thenecessary converse inequality, which confirms the asymptotic sharpnessof the constant in Theorem 2.1 and Strichartz’s Theorem 6.1 [24].

Corollary 2.3 Under the conditions of Theorem 2.1, for any γ ≥ 1,

limλ→∞

λ−γ−2∑

k

(λ− λk )γ+ = Kγ |Ω|

with Kγ as in (2.3).

By a standard Tauberian argument we also have

Corollary 2.4 Under the conditions of Theorem 2.1, for 0 ≤ γ < 1,

limλ→∞

λ−γ−2∑

k

(λ− λk )γ+ =

γ + 3γ + 1

Kγ+1 |Ω|

with Kγ as in (2.3).

Another consequence of Theorem 2.1 follows immediately from theduality

f(x) ≤ g(x), x ≥ 0 ⇔ g(p) ≤ f(p), p ≥ 0,

where the Legendre transform of a convex, non-negative function f isgiven by f(p) = supx≥0(px− f(x)), p ≥ 0. Transforming both sides of(2.1) (cf. [14]) we obtain

Corollary 2.5 Under the conditions of Theorem 2.1 the eigenvalues ofthe Dirichlet Heisenberg Laplacian satisfy the Li-Yau inequality

n∑k=1

λk ≥8√

23|Ω|−1/2n3/2 .


This result constitutes the announced connection with [24]; more pre-cisely, the same inequality follows from combining Theorems 4.2 and 6.1therein.

3 Elliptic case: main results

Now consider the elliptic operator B = −X21 −X2

2 −X23 in L2(R3). For

Ω ⊂ R3 an open set of finite measure we define as before the Dirichletoperator BΩ := PΩBPΩ . In some contrast with Theorem 2.1 we prove

Theorem 3.1 Let Ω ⊂ R3 be a domain of finite measure and let BΩ

be the operator B with Dirichlet boundary conditions in Ω. Then thespectrum of BΩ is discrete and

trϕλ(BΩ) =∑

k

(λ− λk )+ ≤1

15π2 |Ω|λ5/2 .

Remark 3.2 The latter expression coincides which the semi-classicalconstant appearing in the phase-volume Weyl-type asymptotics. Indeed,

limλ→∞

λ−5/2trϕλ (AΩ)

=1

(2π)3 limλ→∞

λ−5/2∫

Ω

∫R3

(λ− a(x, ξ))+ dξ dx =1

15π2 |Ω|,

where a(x, ξ) =(ξ1 + 1

2 x2ξ3)2 +

(ξ2 − 1

2 x1ξ3)2 + ξ2

3 .

4 Hypoelliptic case: results obtained by heat-kerneltechniques

So far we have restricted our study of the Dirichlet Heisenberg LaplacianAΩ to the case where Ω is a domain of finite measure. Put differently,we have studied the Schrödinger-type operator A−V for potential wells,i.e., with V being infinite outside and constant inside Ω. However, us-ing heat-kernel methods we obtain the following upper bound on thecounting function without this restriction on V .

Theorem 4.1 Let N(A − V ) be the number of negative eigenvalues ofA− V . Then for any V ∈ L2(R3).

N(A− V ) ≤ C

∫R3

V (x)2+dx


with

C = mina>0

164a

1e−a + aEi(−a)

≥ 0.09429.

Note that this result is not an improvement in the case of potentialwells; then already Corollary 2.1 gives us the constant 9/28 ≈ 0.03516.However, the asymptotics on the moments of the eigenvalues can beobtained via heat-kernel estimates in conjuction with Karamata’s well-known Tauberian theorem. [10]

Theorem 4.2 Let (λj ) be a non-decreasing sequence of non-negativenumbers such that C = limt→0 tα

∑∞j=0 e−tλj exists for some α > 0.

Then for any f ∈ C([0, 1]),

limt→0

tα∞∑

j=0

f(e−tλj )e−tλj =C

Γ(α)

∫ ∞

0f(e−t)tα−1e−tdt.

In Section 9, in order to prove Theorem 4.1 we derive the equality

e−tA (x, x) =1

32t2,

which implies

tre−tAΩ ≤ 132t2

|Ω|. (4.1)

If the converse inequality also holds – which is still to be proved – so !that

limt→∞

t−2tre−tAΩ =132|Ω|,

then this implies, by Theorem 4.2 with f(s) = s−1χ(1/e,1],

limλ→∞

λ−2N(λ) =1

32Γ(3)|Ω|.

This gives us the asymptotic counterpart of Corollary 2.1, namely

Corollary 4.1 Let N(λ) be the counting function of AΩ , see (2.2). Thenunder the conditions of Theorem 2.1 we have

limt→∞

λ−2N(λ) =164|Ω|.

If we apply Theorem 4.2 with f(s) = λs−1(1+log s)χ(1/e,1] instead (stillassuming (4.1) is true), we get


Corollary 4.2 Under the conditions of Theorem 2.1 we have

limλ→∞

λ−3∑

k

(λ− λk )+ =1

192|Ω|.

As we pointed out after Corollary 2.2, this is a better constant than whatone would have got by straightforward Aizenmann-Lieb extrapolation.

5 Hypoelliptic case: generalisation to R2N +1

The Lie algebra on the Nth Heisenberg group HN is spanned by vectorfields X1 , X2 , . . . , X2N +1 in R2N +1 . If we let

X2k−1 := ∂x2 k −1 +12x2k∂x2 N + 1 , X2k := ∂x2 k

− 12x2k−1∂x2 N + 1

for 1 ≤ k ≤ N and X2N +1 := ∂x2 N + 1 , then the only non-zero commuta-tor is [X2k−1 ,X2k ] = −X2N +1 .

This is the 2N + 1-dimensional version of Theorem 2.1:

Theorem 5.1 Let Ω ⊂ R2N +1 be a domain of finite measure and letAN,Ω , N ≥ 1, be the operator corresponding to the differential expression

−2N∑k=1

X2k

defined in a self-adjoint way in L2(R2) with Dirichlet boundary condi-tions. The spectrum of AN,Ω is discrete and

trϕλ(AN,Ω) ≤ 2cN

(2π)N +1(N + 1)(N + 2)|Ω|λN +2

with

cN :=∑

n1 ,...,nN ≥0

1(2(n1 + . . . + nN ) + N)N +1 . (5.1)

Remark 5.2 The multiple sum (5.1) can be expressed as a single sum,

cN =∞∑

k=0

(N + k − 1

k

)1

(2k + N)N +1 .

Unaware of a closed expression for this number we give approximatevalues for the first constants: c2 ≈ 2.055 · 10−1 , c3 ≈ 2.737 · 10−2 ,c4 ≈ 2.929 · 10−3 , c5 ≈ 2.601 · 10−4 , c6 ≈ 1.969 · 10−5 .


Without hardly modifying the proofs at all, we can extend Corollar-ies 2.1 and 2.2 into

Corollary 5.1 Let N(λ) be the counting function of AN,Ω , see (2.2).Then under the conditions of Theorem 5.1 we have

N(λ) ≤ 2cN (N + 2)N +1

(2π)N +1(N + 1)N +2 |Ω|λN +1 .

Corollary 5.2 Under the conditions of Theorem 5.1 and for any γ > 0the eigenvalues of AN,Ω satisfy the Lieb-Thirring inequality∑

k

(λ− λk )γ+ ≤ KN,γ |Ω|λN +γ+1

with

KN,γ =

2cN (N + 2)N +1

(2π)N +1(N + 1)γγ

(γ + N + 1)γ+N +1 , 0 < γ ≤ 1,

2cN

(2π)N +1(N + 1)(N + 2)6

(γ + 1)(γ + 2), 1 ≤ γ.

6 Proof of Theorems 2.1 and 3.1

Denote by F3 the partial Fourier transform

F3u(x′, ξ3) = (2π)−1/2∫ ∞

−∞e−ix3 ξ3 u(x′, x3) dx3 , x′ = (x1 , x2).

Then,

F3AF∗3 =

(i∂x1 −

12x2ξ3

)2

+(

i∂x2 +12x1ξ3

)2

= (i∇x′ + ξ3A(x′))2 ,

(6.1)where A(x′) = 1

2 (−x2 , x1). This is a Laplacian in x′ with constantmagnetic field ξ3 , on which the Landau levels µn (ξ3) depend:

µn (ξ3) := |ξ3 |(2n + 1), n ∈ N0 = 0, 1, 2, . . ..

Note that F3AF∗3 acts as a multiplication operator with respect to the

variable ξ3 . By the Spectral theorem,

F3AF∗3 =

∞∑n=0

µn (ξ3)Πξ3 ,n , Πξ3 ,n = Π′ξ3 ,n ⊗ IL2 (R) ,


where the projector Π′ξ3 ,n has an explicit representation as an integral

kernel depending on x′ (see, e.g., [9] or [22]), which is constant on thediagonal:

Π′ξ3 ,n (x′, x′) =

|ξ3 |2π

. (6.2)

Let PΩ : L2(R3) → L2(Ω) be the operator of multiplication by χΩ , thecharacteristic function of Ω. The operator AΩ , which is A with Dirichletboundary conditions in Ω, can be identified with the operator PΩAPΩ .By the Berezin-Lieb inequality (see [2], [3], [15] and also [13]), we find

trϕλ(AΩ) ≤trPΩϕλ (A)PΩ

=12π

∫Ω

∫ ∞

−∞

∞∑n=0

ϕλ (µn (ξ3))Πξ3 ,n (x′, x′)dξ3dx

=1

2π2 |Ω|∞∑

n=0

∫ ∞

0(λ− ξ3(2n + 1))+ξ3dξ3 .

Substituting s = ξ3(2n + 1) we have

trϕλ (AΩ) ≤ 12π2 |Ω|

∞∑n=0

1(2n + 1)2

∫ ∞

0(λ− s)+s ds

=1

2π2 |Ω|π2

8λ3

6=

196|Ω|λ3 .

This completes the proof of Theorem 2.1.Similarly to (6.1) we have

F3BF∗3 =F3(−(∂x1 + x2∂x3 )

2 − (∂x2 − x1∂x3 )2 − ∂2

x3)F∗

3

=(i∂x1 + x2ξ3)2 + (i∂x2 − x1ξ3)2 + ξ23

=(i∇x′ + ξ3A(x′))2 + ξ3 .

The eigenvalues of this operator are again functions of ξ3 ,

νn (ξ3) := |ξ3 |(2n + 1) + |ξ3 |2 , n ∈ N0 .

Applying the Berezin-Lieb inequality and recycling some of the compu-tations above we find


trϕλ(BΩ) ≤trPΩϕλ (B)PΩ

=1

(2π)2 |Ω|∫ ∞

−∞|ξ3 |

∞∑k=0

(λ− |ξ3 |(2k + 1)− |ξ3 |2)+ dξ3

≤ 1(2π)2 |Ω|

∫ ∞

−∞

∫ ∞

0|ξ3 |(λ− 2|ξ3 |t− |ξ3 |2)+ dt dξ3

=1

23π2 |Ω|∫ ∞

−∞

∫ ∞

0(λ− t− s2)+ dt ds

=1

24π2 |Ω|∫ ∞

−∞(λ− s2)2

+ ds =1

15π2 |Ω|λ5/2

as claimed.

7 Proof of Corollary 2.1

In order to prove Corollary 2.1 we take up an idea from [12]. Obviously,for any τ > 0,

N(λ) ≤ 1τλ

∑k

((1 + τ)λ− λk )+ ≤196|Ω|λ2 (1 + τ)3

τ. (7.1)

The minimum value of (1 + τ)3τ−1 is reached at τ = 1/2. Substitutingthis value into (7.1) we obtain Corollary 2.1.


The first part of the proof, concerning 0 < γ < 1, is based on thefollowing result obtained in [7]

Lemma 8.1 Let 0 ≤ γ < 1 and λ < µ. Then for all E ≥ 0,

(λ− E)γ+ ≤ C(γ)(µ− λ)γ−1(µ− E)+

with C(γ) := γγ (1− γ)1−γ .

The proof is elementary. Combining this inequality with Theorem 2.1we have, for 0 < γ < 1 and any µ > λ,∑

k

(λ− λk )γ+ ≤C(γ)(µ− λ)γ−1

∑k

(µ− λk )+

≤C(γ)(µ− λ)γ−1 196|Ω|µ3 ≤ 9

32γγ

(γ + 2)γ+2 |Ω|λγ+2


(choose µ = 3λ/(γ + 2)).To determine Kγ for γ > 1 we will use the identity

B(γ − 1, 2)(λ− t)γ+ =

∫ ∞

0(λ− t− µ)+µγ−2dµ, γ > 1,

B being the beta function. Writing∑

(λ− λk )γ+ =

∫(λ− t)γ

+dN(t) andapplying Fubini’s theorem,

B(γ − 1, 2)λ−γ−2∑

k

(λ− λk )γ+ =λ−γ−2

∫ ∞

0µγ−2

∑k

(λ− µ− λk )+dµ

≤K1 |Ω|∫ 1

0s3(1− s)γ−2ds

=K1 |Ω|B(4, γ − 1)

by Theorem 2.1. Taking the supremum of the left-hand side we obtain,by the definition of Kγ ,

Kγ ≤ K1B(4, γ − 1)B(2, γ − 1)

=6

(γ + 1)(γ + 2)K1 =

116(γ + 1)(γ + 2)

, γ > 1.

9 Proof of Theorem 4.1

In this section we assume (U, µ) to be a σ-finite measure space. Anyselfadjoint non-negative operator H in L2(U, µ) generates a contractivesemigroup e−tH t∈R+ . We will assume that the corresponding integralkernels e−tH (x, y) ≥ 0 a.e. in R+ × U × U . Put

MH (t) := ‖e−tH/2‖2L2 →L∞

and suppose this quantity is bounded for all t > 0, MH (t) = O(tα ), α >

0 at zero and integrable at infinity. Moreover, let G(s) be a function onR+ , polynomially growing at infinity and such that G(s)/s is integrableat s = 0. We associate to any such G the function

g(σ) :=∫ ∞

0

G(s)s

e−s/σ ds.

(Note that g(1/σ) is the Laplace transform of G(s)/s.) We shall ap-ply the following bound on the counting function, which is occasionallyreferred to as Lieb’s Formula (see [16] and [25]).

Theorem 9.1 Let H be an operator satisfying the hypotheses above andlet N(H − V ) be the number of negative eigenvalues of H − V . For any


V ∈ L2(U, µ) and any admissible G,

N(H − V ) ≤ 1g(1)

∫ ∞

0

dt

t

∫U

MH (t)G(tV (x))µ(dx). (9.1)

When MH is a homogeneous function, this statement can be simplifiedin the following way. Writing MH (t) = K/tν/2 (and for ease of notationdx instead of µ(dx)) and applying Fubini’s theorem to the right-handside, we find

K

g(1)

∫U

(∫ ∞

0

G(tV (x))tν/2+1 dt

)dx =

K

g(1)

∫ ∞

0

G(s)sν/2+1 ds

∫U

V (x)ν/2dx.

In this case, (9.1) apparently is a (ν-dimensional) Cwikel-Lieb Rozenblyuminequality, whose constant only depends on the power ν.

It is easy to check that the operator A in L2(R3) fits into this frame-work and so N(A − V ) can be estimated using Theorem 9.1. Similarlyto the proof of Theorem 2.1 we calculate, for t > 0, x ∈ R3 ,

e−tA (x, x) = (F∗3 e−tF3 AF∗

3 F3)(x, x)

= (2π)−1∫ ∞

−∞

∞∑n=0

e−µn (ξ3 )tΠξ3 ,n (x, x)dξ3

= π−2∞∑

n=0

∫ ∞

0ξ3e

−2ξ3 (2n+1)tdξ3

=1

(2πt)2

∞∑n=0

1(2n + 1)2

∫ ∞

0se−sds =

132t2

.

This is exactly MA (t) = ess supx∈R3 e−tA (x, x), so ν = 4. One canargue (see [16]) that the optimal G is of the form G(s) = (s − a)+ ,a > 0. Clearly, since∫ ∞

0

G(s)sν/2+1 ds =

12a

and g(1) =∫ ∞

a

(1− a

s

)e−sds = e−a + aEi(−a),

the best constant is the minimum of [64a(e−a + aEi(−a))]−1 , as statedin Theorem 4.1.

10 Proof of Theorem 5.1

Extending our previous definitions we let

F2N +1u(x′, ξ2N +1) = (2π)−1/2∫ ∞

−∞e−ix2 N + 1 ξ2 N + 1 u(x′, x2N +1)dx2N +1 ,


x′ = (x1 , . . . , x2N ) and, for future convenience, x′k = (x2k−1 , x2k ). Then,

F2N +1ANF∗2N +1 =−

N∑k=1

F2N +1(X22k−1 + X2

2k )F∗2N +1

=N∑

k=1

(i∇x′k

+ ξ2N +1A(x′k ))2 ,

where A remains as in Section 6. The eigenvalues of this operator are

µn(ξ2N +1) := |ξ2N +1 |(2|n|+ N), n ∈ NN0 ,

where |n| := n1 + . . . + nN . By the Spectral theorem,

F2N +1ANF∗2N +1 =

∑n∈NN

0

µn(ξ2N +1)Πξ2 N + 1 ,n ,

where Πξ2 N + 1 ,n = Π′ξ2 N + 1 ,n1

⊗ . . . ⊗ Π′ξ2 N + 1 ,nN

⊗ IL2 (R) with Π′ as inSection 6. Clearly,

Πξ2 N + 1 ,n(x′, x′) =|ξ2N +1 |N(2π)N

,

so that

trϕλ(AN,Ω) ≤trPΩϕλ(AN )PΩ

=12π

∫Ω

∫ ∞

−∞

∑n∈NN

0

ϕλ (µn(ξ2N +1))Πξ2 N + 1 ,n(x′, x′)dξ2N +1dx

=2

(2π)N +1 |Ω|∑

n∈NN0

∫ ∞

0(λ− ξ2N +1(2|n|+ N))+ξN

2N +1dξ2N +1

=2

(2π)N +1 |Ω|∑

n∈NN0

1(2|n|+ N)N +1

∫ ∞

0(λ− s)+sN ds

=2

(2π)N +1 |Ω|cNλN +2

(N + 1)(N + 2)

with cN as in (5.1).


The asymptotics stated in the corollary will follow from Corollary 2.2 ifwe can prove the converse bound∑

k

(λ− λk )γ+ ≥ Kγ |Ω|λγ+2 + o(λγ+2), λ →∞, γ ≥ 1. (11.1)


We first assume the domain to be a cylinder, Ω = Ω′× (0, L3). By The-orem 2.1 the spectrum of AΩ consists of eigenvalues λk of finite multi-plicity. If we denote the corresponding L2(Ω)-normalised eigenfunctionsby ωk then by Plancherel’s theorem ‖F3ωk‖2

L2 (Ω′×R) = 2π. As explainedin the proof of Theorem 2.1, the eigenstate of F3AF∗

3 corresponding toµn (ξ3) has infinite dimensionality and is spanned by the eigenfunctions(τξ3 ,m ,n )∞m=−n , which we assume to be normalised in L2(Ω). Evidently,the projection kernel has the form

Π′ξ3 ,n (x′, y′) =

∞∑m=−n

τξ3 ,m ,n (x′)τξ3 ,m ,n (y′).

We finally introduce a cut-off function χε ∈ C∞0 (Ω) such that χε(x) = 1

if dist(x, ∂Ω) ≥ ε for ε > 0, and will use χε(x)eix3 ξ3 τξ3 ,m ,n (x′) as anapproximate eigenfunction.

Setting ϕ(t) := (λ− t)γ+ we have∑

k

ϕ(λk ) =∑

k

ϕ(λk )‖ωk‖22

=12π

∑k

ϕλ (λk )∑m,n

∫ ∞

−∞|(ωk , eix3 ξ3 τξ3 ,m ,n )|2dξ3

≥ 12π

∑k

ϕ(λk )∑m,n

∫ ∞

−∞|(ωk , χεe

ix3 ξ3 τξ3 ,m ,n )|2dξ3

=12π

∫ ∞

−∞

∑m,n

∫ ∞

0ϕ(ν)(dEν χετξ3 ,m ,n , χετξ3 ,m ,n )dξ3 , (11.2)

where Eν is the spectral measure of F3AΩF∗3 . However,∑

m

∫ ∞

0(dEν χετξ3 ,m ,n , χετξ3 ,m ,n )

=∑m

‖χετξ3 ,m ,n‖22 =

∫R3

χ2ε (x)

∑m

|τξ3 ,m ,n (x′)|2dx

=∫

R3χ2

ε (x)Π′ξ3 ,n (x′, x′)dx =

|ξ3 |2π

∫R3

χ2ε (x)dx,

which implies

1− C1ε ≤2π

|ξ3 ||Ω|∑m

∫ ∞

0(dEν χετξ3 ,m ,n , χετξ3 ,m ,n ) ≤ 1− c1ε,

where C1 ≥ c1 > 0 depend on Ω. The convexity of ϕ gives us, by


Jensen’s inequality, the following lower bound to (11.2):

1− C1ε

(2π)2 |Ω|∫ ∞

−∞

∑n

ϕ

(1

1− c1ε

∑m

∫ ∞

0ν(dEν χετξ3 ,m ,n , χετξ3 ,m ,n )

)|ξ3 |dξ3

≥ 1− C1ε

(2π)2 |Ω|∫ ∞

−∞

∑n

ϕ

(µn (ξ3) +

C2ε−2

1− c1ε

)|ξ3 |dξ3

=1− C1ε

16|Ω|

∫ ∞

0

(λ− C2ε

−2

1− c1ε− s

)γ

+s ds

= (1− C1ε)Kγ |Ω|(

λ− C2ε−2

1− c1ε

)γ+2

similarly to the proof of Theorem 2.1. Hence (11.1) holds for cylinders.By the variational principle, the lower bound obtained when we approx-imate an arbitrary domain by decoupled cylinders subject to Dirichletboundary conditions will not be less than the correct one. Since the con-stant actually coincides with that of the upper bound, we have provedthe statement.

Acknowledgements

The authors are grateful for partial support from the ESF EuropeanProgramme “SPECT” and have benefited from the excellent workingenvironment at the Isaac Newton Institute in Cambridge. A. Laptevwould also like to thank Prof. W. Hebisch for useful discussions.

Bibliography[1] M. Aizenmann and E. H. Lieb, On semi-classical bounds for eigenvalues

of Schrödinger operators. Phys. Lett. A 66 (1978) 427–429.[2] F. A. Berezin, Convex functions of operators, Mat. sb. 88 (1972), 268–276.[3] F. A. Berezin, Covariant and contravariant symbols of operators. Math.

USSR Izv. 6 (1972), 1117–1151.[4] M. Sh. Birman and M. Z. Solomyak, The principal term of the spectral

asymptotics formula for “non-smooth" elliptic problems. Functional Anal.Appl., 4 (1970), 265–275.

[5] Z. Ciesielski, On the spectrum of the Laplace operator. Comment. Math.Prace Mat. 14 (1970), 41–50.

[6] L. Erdős, M. Loss and V. Vougalter, Diamagnetic behavior of sums ofDirichlet eigenvalues. Ann. Inst. Fourier (Grenoble) 50 (2000), 891–907.

[7] R. L. Frank, M. Loss and T. Weidl, Eigenvalue estimates of the magneticLaplacian in a domain. In preparation.


[8] L. Hörmander, The Analysis of Linear Partial Differential OperatorsIII. Pseudo-Differential Operators. Springer-Verlag, Berlin-Heidelberg-New York, 1985.

[9] T. Hupfer, H. Leschke and S. Warzel, Upper bounds on the density ofstates of single Landau levels broadened by Gaussian random potentials.J. Math. Phys. 42 (2001), 5626–5641.

[10] J. Karamata, Neuer Beweis und Verallgemeinerung einiger Tauberian-Sätze. Math. Z. 33 (1931), no. 1, 294–299.

[11] P. Kröger, Estimates for sums of eigenvalues of the Laplacian. J. Funct.Anal. 126 (1994), 217–227.

[12] A. Laptev, Dirichlet and Neumann Eigenvalue Problems on Domains inEuclidean Spaces. J. Func. Anal. 151 (1997), 531–545.

[13] A. Laptev and Yu. Safarov, A generalization of the Berezin-Lieb inequal-ity. Amer. Math. Soc. Transl (2) 175 (1996), 69–79.

[14] A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities.Journées "Équations aux Dérivées Partielles" (La Chapelle sur Erdre,2000), Exp. No. XX, Univ. Nantes, Nantes (2000).

[15] E. H. Lieb, The classical limit of quantum spin systems. Comm. Math.Phys. 31 (1973), 327–340.

[16] E. H. Lieb, The number of bound states of one-body Schrödinger operatorsand the Weyl problem. Proc. Sym. Pure Math. XXXVI, 241–252. Amer.Math. Soc., Providence RI, 1980.

[17] E. H. Lieb and W. Thirring, Inequalities for the moments of the eigen-values of the Schrödinger Hamiltonian and their relation to Sobolev in-equalities. Studies in Mathematical Physics, Essays in Honor of ValentineBargmann, 269–303. Princeton University Press, Princeton NJ, 1976.

[18] P. Li and S.-T. Yau, On the Schrödinger equation and the eigenvalueproblem. Comm. Math. Phys. 88 (1983), 309–318.

[19] G. Métivier, Fonction spectrale et valeurs propres d’une classed’opérateurs non elliptiques. Comm. Partial Differential Equations 1(1976), 467–519.

[20] G. Métivier, Valeurs propres de problèmes aux limites elliptiques ir-réguliers. Bull. Soc. Math. France, Mem. 51–52 (1977), 125–229.

[21] G. Pólya, On the eigenvalues of vibrating membranes. Proc. LondonMath. Soc. 11 (1961), 419–433.

[22] G. D. Raikov and S. Warzel, Quasi-classical versus non-classical spectralasymptotics for magnetic Schrödinger operators with decreasing electricpotentials. Rev. Math. Phys. 14 (2002), no. 10, 1051–1072.

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6Equidistribution for quadratic differentials

Ursula HamenstädtMathematisches Institut

Rheinische Friedrich-Wilhelms-Universität [email protected]

1 Introduction

Hermann Weyl was one of the most influential mathematicians of thefirst half of the twentieth century. He was born in 1885 in Elmshorn. In1933 he emigrated to the United States, and he died in 1955 in Zürich.The majority of his many fundamental contributions to mathematicsbelong to the area of analysis in the broadest possible sense. However,one of his earliest and most celebrated results can be viewed as the originof the study of number theory with tools from dynamical systems.

His theorem, published in 1916 in the “Mathematische Annalen” [W16],is as follows.

Weyl’s theorem: Let y0 ∈ (0, 1) be irrational. Then the sequence(ui)i≥1 defined by ui = iy0 mod 1 is asymptotically equidistributed: Forall 0 < a < b < 1 we have

|1 ≤ i ≤ n : a ≤ ui ≤ b|n

→ b− a (n →∞).

In this note we explain how the idea behind this theorem was used inthe last quarter of the twentieth century to gain surprising insights intothe interplay between number theory, geometry and dynamical systems.

2 Classical dynamical systems

In this section we discuss how Weyl’s theorem can be reformulated in thelanguage of dynamical systems, and we introduce some basic conceptswhich will be important in the later sections.

Let S1 = eit | t ∈ [0, 2π) ⊂ C be the standard unit circle in thecomplex plane. Then S1 is an abelian group with multiplication eit ×

116

Equidistribution for quadratic differentials 117

eis = ei(t+s) . In particular, every angle α ∈ (0, 2π) defines a cyclic groupTn

α of rotations of S1 (n ∈ Z) via

Tnα (eit) = eit+nα .

Thus the rotation Tα with angle α generates a dynamical system withphase space S1 . Global (or asymptotic) properties of a dynamical systemcan be investigated with the help of invariant measures.

Definition 2 A Radon measure µ (i.e. a locally finite Borel measure)on a locally compact topological space X is invariant under a Borel mapT if µ(T−1(A)) = µ(A) for every Borel set A ⊂ X.

If X is a compact topological space then the space P(X) of Borel prob-ability measures on X can be equipped with the weak∗-topology. Thisweak∗- topology is the weakest topology such that for every continuousfunction f : X → R the function µ ∈ P(X) →

∫fdµ is continuous. In

other words, a sequence (µi) ⊂ P(X) converges to µ ∈ P(X) if and onlyif for every open subset U of X we have lim inf i→∞ µi(U) ≥ µ(U). Thespace P(X) equipped with the weak∗-topology is compact.

By compactness, every continuous transformation T of X admits aninvariant Borel probability measure [Wa82]. Namely, for every pointx ∈ X, any weak limit of the sequence of measures

1n

n−1∑i=0

δT i x

is T -invariant where δz is the Dirac δ-measure at z, defined by δz (z) =1 and δz (X − z) = 0.

For our circle rotations, there are now two cases.Case 1: α is a rational multiple of 2π, i.e α = 2pπ/q for relatively

prime p, q ∈ N.In this case we have T q

α (eit) = eit+2pπ = eit for all t which means thefollowing.

Every point in S1 is periodic for Tα , with period independent of the point.

In particular, every point y ∈ S1 is an atom of a Tα -invariant prob-ability measure, namely the weighted counting measure on the orbit

118 Ursula Hamenstädt

T iαy | 0 ≤ i ≤ q − 1 of y. This measure is given by the formula

µ =1q

q−1∑i=0

δT iα y .

Case 2: α is an irrational multiple of 2π, i.e. α = 2πρ for an irrationalnumber ρ ∈ (0, 1).

In this case, Tα does not have periodic points, and Weyl’s theoremsays precisely the following: For each y ∈ S1 ,

1n

n−1∑i=0

δT iα y → λ

weakly in the space of probability measures on S1 where λ is the nor-malized standard Lebesgue measure on S1 defined by λeis | 0 ≤ α <

s < β ≤ 2π = (β − α)/2π.For a continuous map T of a compact space X, the space P(X)T of

T -invariant Borel probability measures on X is convex: If µ1 , µ2 are twosuch measures and if s ∈ [0, 1] then sµ1 + (1 − s)µ2 ∈ P(X)T as well.In other words, P(X)T is a compact and convex subset of a topologicalvector space on which the dual separates points. Hence P(X)T is theconvex hull of the set of its extreme points.

An extreme point µ ∈ P(X)T is an ergodic invariant measure: IfA ⊂ X is a T -invariant Borel set then µ(A) = 0 or µ(X − A) = 0. Bythe Birkhoff ergodic theorem [Wa82], every extreme point µ ∈ P(X)T isa weak limit of measures of the form 1

q

∑q−1i=0 δT i y for a suitable choice

of y ∈ X. Note that the definition of ergodicity also makes sense forRadon measures on locally compact spaces which are invariant under acontinuous transformation.

Definition 3 A continuous transformation T of a compact space X iscalled uniquely ergodic if T admits a unique invariant Borel probabilitymeasure.

An invariant Borel probability measure µ for a uniquely ergodic con-tinuous transformation T of a compact space X is necessarily ergodic.Now Weyl’s theorem can be rephrased as follows.

Irrational rotations of the circle are uniquely ergodic.

However, this also means the following.

If α is an irrational multiple of 2π then a measure which is invariantunder Tα is invariant under the full circle group of rotations.


3 The modular group and hyperbolic geometry

About 1970, the significance of Weyl’s theorem became apparent in asomewhat unexpected way and in a different context. This developmentbegan with the work of Hillel Furstenberg. Furstenberg was born in1935 in Berlin and moved shortly later with his family to the UnitedStates. He now works at the Hebrew University in Jerusalem (Israel).Furstenberg was interested in lattices in semi-simple Lie groups G ofnon-compact type and their actions on homogeneous spaces associatedto G. A large part of the structure theory for semi-simple Lie groupsis due to Hermann Weyl, but it seems that he never attempted to drawa close connection between the structure of Lie groups, their actions onhomogeneous spaces and his number theoretic result which we discussedin Section 1.

In this section we explain Furstenberg’s work and its generalizationswhich are entirely in the spirit of Weyl’s theorem.

Consider the modular group

SL(2, Z) = (

a b

c d

)| a, b, c, d ∈ Z, ad− bc = 1

which acts as a group of linear transformations on R2 preserving theusual area form. This action is given by(

a b

c d

)(x

y

)→

(ax + by

cx + dy

).

There is an obvious SL(2, Z)-invariant subset of R2 , namely the count-able set RQ2 ⊂ R2 of points whose coordinates are dependent over Q(which means that their quotient is rational). Since SL(2, Z) preservesthe integral lattice Z2 ⊂ R2 , each SL(2, Z)-orbit of a point whose coordi-nates are dependent over Q is a discrete subset of R2 . Hence this orbitsupports an SL(2, Z)-invariant purely atomic ergodic Radon measure.For example, the measure

µ =∑y∈Z2

δy

is an SL(2, Z)-invariant Radon measure. However, it is not ergodic sinceZ2 contains countably many orbits for the action of SL(2, Z). Namely,the SL(2, Z)-orbit of the point (1, 0) ∈ Z2 consists precisely of all points(p, q) such that p, q ∈ Z are relatively prime. As a consequence, thereis an uncountable family of SL(2, Z)-invariant Radon measures on R2 .


Each ergodic measure in this family is a sum of weighted Dirac masseson a single SL(2, R)-orbit in RQ2 .

In contrast, extending earlier work of Furstenberg [F72], Dani [D78]proved in 1978 the following unique ergodicity result.

Theorem 3.1 (Unique ergodicity for the standard linear actionof SL(2, Z)):An SL(2, Z)-invariant Radon measure on R2 which gives full mass to theset of points whose coordinates are independent over Q coincides withthe Lebesgue measure up to scale.

As a consequence, we have.

A Radon measure on R2 which is invariant under SL(2, Z) and whichgives full measure to points whose coordinates are independent over Q

is invariant under the full group SL(2, R).

The proof of this result does not use directly the fact that SL(2, Z) actson R2 by linear transformations. Instead, the group SL(2, Z) is viewed asa lattice in the simple Lie group SL(2, R). By this we mean that SL(2, Z)is a discrete subgroup of SL(2, R) with the following property. The groupSL(2, R) admits a natural Radon measure which is invariant under theaction of SL(2, R) on itself by right or left translation. This measure isgiven by a biinvariant volume form. By biinvariance, this volume formprojects to a volume form on the quotient orbifold SL(2, Z)\SL(2, R) offinite total volume.

The quotient group PSL(2, R) under the center Z/2Z of SL(2, R)admits a natural simply transitive action on the unit tangent bundleT 1H2 of the hyperbolic plane H2 and hence this unit tangent bundlecan be identified with PSL(2, R). Namely, we have H2 = z = x + iy ∈C | Im(z) > 0 with the Riemannian metric

Q =dx2 + dy2

y2

which is invariant under the action of SL(2, R) by linear fractional trans-formations

z → az + b

cz + dwhere

(a b

c d

)∈ SL(2, R).

The subgroup of SL(2, R) acting trivially is just the center of SL(2, R)and hence this action factors to an action of PSL(2, R). The hyperbolicplane H2 admits a compactification by adding the circle ∂H2 = R ∪∞,


and the action of PSL(2, R) on H2 extends to a transitive action on thiscircle by homeomorphisms.

There are three characteristic one-parameter subgroups of SL(2, R).

(i) The diagonal subgroup

A = (

et 00 e−t

)| t ∈ R

(ii) The upper unipotent group

N = (

1 t

0 1

)| t ∈ R

(iii) The lower unipotent group

U = (

1 0t 1

)| t ∈ R

These groups project to one-parameter subgroups of PSL(2, R) whichwe denote by the same symbols.

The right action of the diagonal subgroup A on PSL(2, R) definesthe geodesic flow on T 1H2 . The right action of the group N of uppertriangular matrices of trace two is the horocycle flow on T 1H2 . Thegroup PSL(2, R) acts transitively from the left on the homogeneous spacePSL(2, R)/N .

Recall that the linear action of SL(2, R) on R2 naturally induces anaction of PSL(2, R) on the punctured cone R2 − 0/± 1. We have.

Lemma 3.1 There is a homeomorphism

F : R2 − 0/± 1 → PSL(2, R)/N

which commutes with the action of PSL(2, R). This means that we haveB(Fz) = F (Bz) for all z ∈ R2 − 0/± 1 and for all B ∈ PSL(2, R).

Proof A homeomorphism as required in the lemma can easily be de-termined explicitly (see e.g. the paper [LP03]). However, its existencecan also be derived as follows. The group PSL(2, R) acts transitivelyfrom the left on R2 − 0/ ± 1 (this is immediate from transitivity ofthe left linear action of SL(2, R) on R2 − 0). Moreover, the stabilizersubgroup of the point (1, 0)/±1 for this action is precisely the group N .


As a consequence, PSL(2, Z)-invariant Radon measures on R2−0/±1correspond precisely to Radon measures on PSL(2, R)/N which are in-variant under the left action of PSL(2, Z) or, equivalently, to finite Borelmeasures on the unit tangent bundle T 1(Mod) = PSL(2, Z)\PSL(2, R)of the modular surface Mod = PSL(2, Z)\H2 which are invariant underthe action of the horocycle flow ht defined by the right action of theupper unipotent group N .

There is an obvious family of ht-invariant Borel probability measureson the homogeneous space T 1(Mod). Namely, a fundamental domain forthe action of PSL(2, Z) on the hyperbolic plane H2 by linear fractionaltransformations is the complement of the euclidean disc of radius onecentered at the origin in the strip z ∈ C | Im(z) > 0, 1

2 ≤ Re(z) ≤ 12 .

The stabilizer of ∞ in the group PSL(2, R) is the solvable subgroup G

of all upper triangular matrices generated by A and N . This stabilizeris preserved by the action of N by right translation. The orbits of N inS project to the lines Im = const in H2 and hence they project to closedorbits of the horocycle flow on T 1(Mod). In particular, for every suchorbit there is a unique ht-invariant Borel probability measure supportedon this orbit. Figure 1 shows a periodic orbit of the horocycle flow aboutthe cusp in the standard fundamental domain of the action of the groupPSL(2, Z) on H2 .

1.5

0.5−0.5

0.5

1

2

To describe the corresponding PSL(2, Z)-invariant Radon measureon the cone R2 − 0/ ± 1, observe that the left action of PSL(2, R)on the cone R2 − 0/ ± 1 projects to the action of PSL(2, R) on thereal projective line RP 1 ∼ S1 by projective transformations. This ac-tion is transitive, and the stabilizer of the real line [1, 0] spanned by


the point (1, 0) ∈ R2 equals the subgroup G of PSL(2, R). Thus theaction of PSL(2, R) on ∂H2 is just the action of PSL(2, R) on RP 1 .In particular, the PSL(2, Z)-orbit of [1, 0] which consists of all pointsin RP 1 spanned by vectors with integer coordinates (see the abovediscussion) naturally coincides with the PSL(2, Z)-orbit of the point∞ ∈ ∂H2 . As a consequence, the ht-invariant Borel probability mea-sures on PSL(2, Z)\PSL(2, R) supported on the above described closedorbits of the horocycle flow correspond precisely to the invariant Radonmeasures on R2 − 0/ ± 1 supported on points whose coordinates aredependent over Q. Thus we have.

Borel probability measures supported on closed orbits of ht on T 1 (Mod)correspond to SL(2, Z)-invariant Radon measures on R2 supported onorbits of points whose coordinates are dependent over Q.

On the other hand, the Lebesgue Haar measure is invariant under theaction of the whole group PSL(2, R), and it is uniquely determined bythis property up to scale. Thus Theorem 3.1 is an immediate conse-quence of the following result of Dani [D78].

Proposition 3.2 Any ht-invariant probability measure on T 1(Mod) ei-ther is supported on a closed orbit for the horocycle flow or it is invariantunder the whole group PSL(2, R).

In the early nineties, Ratner proved a far-reaching generalization ofthis result. We refer to the book [BM00] for an introduction to the sub-ject and to [WM05] for a more detailed treatment of Ratner’s celebratedwork.

4 Uniquely ergodic unipotent flows on some homogeneousspaces of infinite volume

The results explained in Section 3 and their generalizations, in particularthe work of Ratner, have many applications. However, they are onlyapplicable in an algebraic setting and to invariant probability measures.The simplest extension of the questions discussed in Section 3 which cannot be answered with these methods can be formulated as follows.

Let S be a closed oriented surface of genus g ≥ 2. Choose a Rieman-nian metric g on S of constant sectional curvature −1. Then there isa discrete subgroup Γ of PSL(2, R) such that our hyperbolic surface isjust Γ\H2 , with unit tangent bundle T 1S = Γ\PSL(2, R) as before. In


particular, the horocycle flow ht is defined on T 1S. For some d ≤ 2g

choose a normal subgroup Λ of Γ with factor group Γ/Λ isomorphic toZd . An example of such a group is the commutator subgroup of Γ. Con-sider the regular Zd -cover S of S with fundamental group Λ. Then thehorocycle flow ht on the unit tangent bundle T 1 S of S is defined. By thework of Ratner, ht-invariant Borel probability measures on T 1 S can beclassified. Namely, in the situation at hand, either they are supportedon closed orbits of the horocycle flow or they are invariant under the fullgroup PSL(2, R) on T 1 S. In other words, since the volume of T 1 S isinfiniteand since there are no closed orbits for the horocycle flow, such in-variant Borel probability measures do not exist. However, the Lebesguemeasure (i.e. the measure induced by a Haar measure on PSL(2, R))is a PSL(2, R)-invariant Radon measure on T 1 S. A natural problem isnow to classify all invariant Radon measures for the horocycle flow onT 1 S.

Babillot and Ledrappier [BL98] constructed for every homomorphismϕ : Zd → R a Radon measures λϕ on T 1 S which is both invariant un-der the horocycle flow on T 1 S and under the geodesic flow. The trivialhomomorphism corresponds to the Lebesgue measure. Each of thesemeasures is the lift of a Borel probability measure λϕ on T 1S. If the ho-momorphism is nontrivial, then the measure λϕ on T 1S is not invariantunder the horocycle flow on T 1S. For ϕ = ψ the measures λϕ , λψ aresingular. We call these measures Babillot-Ledrappier measures.

The Babillot-Ledrappier measures are all absolutely continuous withrespect to the stable foliation, i.e. the foliation of T 1 S = Λ\PSL(2, R)into the orbits of the right action of the solvable subgroup G generatedby the groups A,N . More precisely, the following holds true.

For every point ξ ∈ S1 = ∂H2 there is a Busemann function θξ :H2 → R at ξ. Such a Busemann function is a one-Lipschitz functionfor the hyperbolic metric on H2 which is determined uniquely by ξ upto an additive constant. The function θ∞(z) = log Im(z) is a Busemannfunction at the point ∞. The Busemann functions are invariant underthe action of PSL(2, R) on ∂H2 ×H2 and therefore the images underthe action of PSL(2, R) of the function θ∞ determines all Busemannfunctions on H2 .

For a discrete subgroup Λ of PSL(2, R) and a number α ≥ 0 define aconformal density of dimension α ≥ 0 for Λ to be an assignment whichassociates to every x ∈ H2 a finite measure µx on ∂H2 with the followingproperties.


(i) The measures µx (x ∈ H2) are equivariant under the action of Λon H2 × ∂H2 .

(ii) For x, y ∈ H2 the measures µx, µy are absolutely continuous, withRadon Nikodym derivative dµy

dµx (ξ) = eα(θξ (y )−θξ (x)) where θξ is aBusemann function at ξ.

The Babillot-Ledrappier measures on T 1 S are related to conformal den-sities for the fundamental group Λ of S as follows.

Recall that there is a Λ-equivariant canonical projection PSL(2, R) →∂H2 = PSL(2, R)/G. Let µ be the Λ-invariant measure on PSL(2, R)which is the lift of a Babillot-Ledrappier measure λϕ on

T 1 S = Λ\PSL(2, R).

Then for every relatively compact open subset U of PSL(2, R) the push-forward π∗(µ|U) is contained in the measure class of a conformal densityfor Λ. This conformal density µx has the additional property that

µx g = eϕ(g) µx∀g ∈ Zd .

This additional condition determines the conformal density uniquely upto scale. The measure λϕ in turn is uniquely determined by the confor-mal density up to scale.

A Radon measure µ on T 1 S is quasi-invariant under the geodesic flowΦt on T 1 S if for every t ∈ R the push-forward measure Φt

∗µ is absolutelycontinuous with respect to µ, i.e. the measures µ and Φt

∗µ have the samesets of measure zero. The following result is due to Babillot [B04] andAaronson, Nakada, Sarig, Solomyak [ASS02].

Proposition 4.1 Every ht-invariant Radon measure on T 1 S which isquasi-invariant under the geodesic flow is a Babillot-Ledrappier measure.

Now let λ by any ht-invariant ergodic Radon measure on T 1 S. LetΦt be the geodesic flow on T 1 S and let H(λ) ⊂ R be the set of allt ∈ R such that the measure Φtλ is contained in the measure class ofλ. Then H(λ) is a closed subgroup of R and hence if H(λ) = R theneither H(λ) is infinite cyclic or trivial. The measure λ is quasi-invariantunder the flow Φt if and only if we have H(λ) = R. Proposition 4.1then shows that every ht-invariant Radon measure λ with H(λ) = R isa Babillot-Ledrappier measure.

To classify the ht-invariant measures with the property that H(λ) iseither infinite cyclic or trivial, Sarig [S04] proved a general structure


theorem for cocycles. He uses this result to show that there is no ht-invariant Radon measure λ on T 1 S with Hλ = R. Thus he obtains.

Theorem 4.1 Every ht-invariant ergodic Radon measure on T 1 S is aBabillot-Ledrappier measure.

In fact, the analog of Theorem 4.1 holds true for the horocycle flowon any Zd -cover of a closed surface S of higher genus equipped with aRiemannian metric of negative Gauß curvature. In other words, thisresult does not require any algebraic setting.

5 Moduli space

The first book written by Hermann Weyl is the monograph “Die Ideeder Riemannschen Fläche” which appeared in 1913. In this section, aRiemann surface will be a closed oriented surface of genus g ≥ 1 whichis equipped with a complex structure. We are going to connect themoduli space of Riemann surfaces to the ideas discussed in Section 3and Section 4.

Define a marked Riemann surface to be a Riemann surface M togetherwith a homeomorphism S → M from a fixed closed oriented surface S

of genus g ≥ 1.

Definition 4

(i) The Teichmüller space T (S) of S is the space of all marked Rie-mann surfaces which are homeomorphic to S up to biholomor-phisms isotopic to the identity.

(ii) The mapping class group M(S) is the group of all isotopy classesof orientation preserving homeomorphisms of S.

Every element of the mapping class group M(S) naturally induces anontrivial automorphism of the fundamental group π1(S) of S. It is easyto see that this automorphism is not inner, i.e. it is not induced by aconjugation. Thus there is a natural homomorphism of M(S) into thegroup Out(π1(S)) of outer automorphisms of π1(S), i.e. the quotient ofthe group of all automorphisms of π1(S) by the normal subgroup of allinner automorphisms. By an old result of Nielsen, this map is in fact anisomorphism.

The mapping class group naturally acts on Teichmüller space by pre-composition, i.e. by changing the markings. It is well known that T (S)can be equipped with a topology so that with respect to this topology,


T (S) is homeomorphic to R6g−6 and that the mapping class group actsproperly discontinuously on T (S) by homeomorphisms. There is also anM(S)-invariant complex structure on T (S) which identifies T (S) witha bounded domain Ω in C3g−3 . The mapping class group is then justthe group of all biholomorphic automorphisms of Ω.

By uniformization, if g ≥ 2 then the moduli space

Mod(S) =M(S)\T (S)

of S can be identified with the space of all hyperbolic Riemannian metricson S up to orientation preserving isometries. In the case g = 1, it is thespace of all euclidean metrics of area one up to orientation preservingisometries.

Example:In the case g = 1 (i.e. in the case of the 2-torus S = T 2), the fun-

damental group π1(S) of S is the lattice Z2 in R2 . Then every auto-morphism of Z2 is induced by a linear isomorphism of R2 preserving thelattice Z2 and hence the mapping class group M(S) is just the groupSL(2, Z). Here the center Z/2Z of SL(2, Z) corresponds to the hyperel-liptic involution which acts trivially on T (T 2). More precisely, we havenatural identifications as follows.

(i) T (T 2) = H2 = z ∈ C | Im(z) > 0, the hyperbolic plane.(ii) M(T 2) = SL(2, Z) acting on H2 by linear fractional transforma-

tions.(iii) Mod(T 2) = SL(2, Z)\H2 , the modular surface.

A Riemann surface S is a one-dimensional complex manifold and henceit admits a natural holomorphic cotangent bundle T ′(S) whose fiber ata point x is the one-dimensional C-vector space of all C-linear mapsTxS → C.

Definition 5 For a Riemann surface S, a holomorphic quadratic dif-ferential q on S is a holomorphic section of the holomorphic line bundleT ′(S)⊗ T ′(S).

In a holomorphic coordinate z on S, a holomorphic quadratic differen-tial q can be written in the form q(z) = f(z)dz2 with a local holomorphicfunction f . The bundle of all holomorphic quadratic differentials over allRiemann surfaces can be viewed as the cotangent bundle of Teichmüllerspace. It is a complex vector bundle of complex dimension 3g − 3. The


mapping class group M(S) acts properly discontinuously on this bundleas a group of bundle automorphisms.

A quadratic differential q defines a singular euclidean metric on S asfollows. Near a regular point z, i.e. away from the zeros of the differen-tial, there is a holomorphic coordinate z on S such that in this coordi-nate the differential is just dz2 . Such a chart is unique up to translationand multiplication with −1 and hence the euclidean metric defined bythis chart is uniquely determined by q. We call such a chart isometric.The area of a quadratic differential is the area of the singular euclideanmetric it defines. The mapping class group preserves the sphere bundleQ(S) over T (S) of all holomorphic quadratic differentials of area oneand hence this bundle projects to the moduli space Q(S) = M(S)\Q(S)of such differentials.

The real line bundles q > 0, q < 0 on the complement of the (finitelymany) singular points of q define transverse singular measured foliationsqh , qv on S called the horizontal and vertical measured foliations of q.By definition, a measured foliation of S is a foliation F with finitelymany singularities together with a transverse measure which associatesto every smooth compact arc which meets the leaves of the foliation F

transversely a length which is invariant under a homotopy of the arcmoving each endpoint of the arc within a single leaf of the foliation.

There is a natural action of the group SL(2, R) on the space Q(S) ofarea one holomorphic quadratic differentials which is given as follows.For each quadratic differential q ∈ Q(S) choose a family of isometriccharts near the regular points. For B ∈ SL(2, R) define Bq to be thequadratic differential whose isometric charts are the compositions of theisometric charts for q with B. This collection of charts then defines a newholomorphic quadratic differential on a (different) Riemann surface. Theaction of SL(2, R) commutes with the action of the mapping class groupand hence it descends to an action of SL(2, R) on Q(S). The diagonalsubgroup of SL(2, R) then defines a flow on Q(S) called the Teichmüllerflow Φt , and the upper unipotent group defines the horocycle flow ht . Inthe case of the Teichmüller space of surfaces of genus 1, these flows areprecisely the geodesic flow and the horocycle flow on the unit tangentbundle of the modular surface. By construction, the horocycle flowpreserves the horizontal measured foliation of the quadratic differentialssince it preserves the lines q > 0 in our charts.

For the moduli space of surfaces of higher genus, the classification ofht-invariant Borel probability measures is up to date impossible. Thereare lots of examples of such measures. For example, Veech surfaces in


moduli space are holomorphically embedded (singular) Riemann surfacesof finite type. They correspond to closed SL(2, R)-orbits in Q(S). Anyht-invariant Borel probability measure on such an orbit then defines aht-invariant Borel probability measure on Q(S).

However, we can ask for the easier question of a classification ofM(S)-invariant Radon measures on the space of equivalence classes of mea-sured foliations. For this call two measured foliations on S are equivalentif they can be transformed into each other by so-called collapses of twosingular points along a connecting compact singular arc and Whiteheadmoves. Figure 2 shows a modification of a singular foliation with such aWhitehead move.

The following fundamental fact was discovered by Hubbard and Masurin 1979 [HM79], see also [Hu06].

Theorem 5.1 (Hubbard-Masur) Let H be the natural map which asso-ciates to a quadratic differential the equivalence class of its horizontalmeasured foliation. Then for every Riemann surface M the restrictionof H to the truncated vector space of all nontrivial quadratic differentialson M is a bijection.

Example: If S = T 2 then the space of equivalence classes of measuredfoliations is R2 .

By the theorem of Hubbard and Masur (in a slightly stronger versionthan the one we described above), the natural topology on the bun-dle Q(S) induces a metrizable topology on the set MF of equivalenceclasses of measured foliations on S (which by construction is a purelytopologically defined space). The mapping class groupM(S) acts on thespace of equivalence classes of measured foliations by homeomorphisms.If S = T 2 then this action can naturally be identified with the linearaction of SL(2, R) on R2 .


Definition 6 A measured foliation F on S fills up S if there is no simpleclosed curve on S whose F -length vanishes.

Here the F -length of a simple closed curve c is the infimum of thetransverse lengths of closed curves transverse to the foliation which arefreely homotopic to c.

The following classification result which was independently and atthe same time shown in [H07a] and in [LM07] extends unique ergodicityfor the action of SL(2, Z) on the set of irrational points in R2 to theframework of Teichmüller theory and moduli spaces. For the formulationof this result, define a measured multi-cylinder for a measured foliationof S to be a disjoint union of embedded annuli in S which are foliatedby closed leaves of the foliation.

Theorem 5.2 Let µ be an M(S)-invariant ergodic Radon measure onMF .

(i) If µ gives full mass to measured foliations which fill up S then µ

coincides with the Lebesgue measure up to scale.(ii) If µ is singular to the Lebesgue measure then there is a mea-

sured foliation F containing a nontrivial measured multi-cylinderc such that µ coincides with the translates of a Stab(c)-invariantmeasure on the space of measured foliations on S − c.

Remark: The proof for genus g ≥ 2 is not valid in the case g = 1, i.e.we do not obtain a new proof of the result of Dani. The argument forthe first part of the theorem uses the structural result of Sarig discussedin Section 4 in an essential way. The proof of the second part relies on aresult of Minsky and Weiss [MW02] which is motivated by an analogousclassical result of Dani for the horocycle flow on non-compact hyperbolicsurfaces of finite volume.

Finally we discuss some applications. We begin with two classicalresults of Margulis [M69] and Dani.

Theorem 5.3

(i) (Margulis) For > 0 let n() be the number of closed geodesicson M = SL(2, Z)\H2 of length at most . Then

lim→∞

1

log n() = 1.

(ii) (Dani) For a compact subset K of SL(2, Z)\H2 and for > 0 let


nK () be the number of periodic orbits of length at most whichare entirely contained in K. Then

1 = suplim inf→∞

1

log nK () | K ⊂ SL(2, Z)\H2compact .

Also, for a hyperbolic surface M a collar lemma holds: There is acompact subset K in M such that every geodesic in M intersects K.

For closed Teichmüller geodesics, Eskin and Mirzakhani [EM08] ob-tained recently the analog of Margulis’ result.

Theorem 5.4 Let n() be the number of closed Teichmüller geodesicsof length at most . Then

lim→∞

1

log n() = 6g − 6.

We also have [H07b].

Theorem 5.5 For a compact subset K of Mod(S) and for > 0 letnK () be the number of closed geodesics in Mod(S) which are entirelycontained in K; then

6g − 6 = suplim inf→∞

1

log nK () | K ⊂ Mod(S) compact .

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spaces, J. reine angew. Math. 552 (2002), 131-177.[S04] O. Sarig, Invariant Radon measures for horocycle flows on Abelian cov-

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versity Press, Chicago 2005.

7Weyl’s law in the theoryof automorphic forms

Werner MüllerUniversität Bonn

Mathematisches [email protected]

1 Introduction

Let M be a smooth, compact Riemannian manifold of dimension n withsmooth boundary ∂M (which may be empty). Let

∆ = −div grad = d∗d

be the Laplace-Beltrami operator associated with the metric g of M .We consider the Dirichlet eigenvalue problem

∆ϕ = λϕ, ϕ∣∣∂M

= 0. (1.1)

As is well known, (1.1) has a discrete set of solutions

0 ≤ λ0 ≤ λ1 ≤ · · · → ∞

whose only accumulation point is at infinity and each eigenvalue oc-curs with finite multiplicity. The corresponding eigenfunctions ϕi canbe chosen such that ϕii∈N0 is an orthonormal basis of L2(M). A fun-damental problem in analysis on manifolds is to study the distributionof the eigenvalues of ∆ and their relation to the geometric and topo-logical structure of the underlying manifold. One of the first results inthis context is Weyl’s law for the asymptotic behavior of the eigenvaluecounting function. For λ ≥ 0 let

N(λ) = #j :

√λj ≤ λ

be the counting function of the eigenvalues of

√∆, where eigenvalues

are counted with multiplicities. Denote by Γ(s) the Gamma function.Then the Weyl law states

N(λ) =vol(M)

(4π)n/2Γ(

n2 + 1

)λn + o(λn ), λ →∞. (1.2)

133

134 Werner Müller

This was first proved by Weyl [We1] for a bounded domain Ω ⊂ R3 .Written in a slightly different form it is known in physics as the Rayleigh-Jeans law. Raleigh [Ra] derived it for a cube. Garding [Ga] provedWeyl’s law for a general elliptic operator on a domain in Rn . For aclosed Riemannian manifold (1.2) was proved by Minakshisundaram andPleijel [MP].

Formula (1.2) does not say very much about the finer structure ofthe eigenvalue distribution. The basic question is the estimation of theremainder term

R(λ) := N(λ)− vol(M)(4π)n/2Γ

(n2 + 1

)λn .

That this is a deep problem shows the following example. Consider theflat 2-dimensional torus T = R2/(2πZ)2 . Then the eigenvalues of theflat Laplacian are λm,n := m2 + n2 , m,n ∈ Z and the counting functionequals

N(λ) = #(m,n) ∈ Z2 :

√m2 + n2 ≤ λ

.

Thus N(λ) is the number of lattice points in the circle of radius λ. Anelementary packing argument, attributed to Gauss, gives

N(λ) = πλ2 + O (λ) ,

and the circle problem is to find the best exponent µ such that

N(λ) = πλ2 + Oε

(λµ+ε

), ∀ε > 0.

The conjecture of Hardy is µ = 1/2. The first nontrivial result is due toSierpinski who showed that one can take µ = 2/3. Currently the bestknown result is µ = 131/208 ≈ 0.629 which is due to Huxley. Levitan[Le] has shown that for a domain in Rn the remainder term is of orderO(λn−1).

For a closed Riemannian manifold, Avakumović [Av] proved the Weylestimate with optimal remainder term:

N(λ) =vol(M)

(4π)n/2Γ(

n2 + 1

)λn + O(λn−1), λ →∞. (1.3)

This result was extended to more general, and higher order operatorsby Hörmander [Ho]. As shown by Avakumović the bound O(λn−1) ofthe remainder term is optimal for the sphere. On the other hand, undercertain assumption on the geodesic flow, the estimate can be slightlyimproved. Let S∗M be the unit cotangent bundle and let Φt be thegeodesic flow. Suppose that the set of (x, ξ) ∈ S∗M such that Φt has a

Weyl’s law in the theory of automorphic forms 135

contact of infinite order with the identity at (x, ξ) for some t = 0, hasmeasure zero in S∗M . Then Duistermaat and Guillemin [DG] provedthat the remainder term satisfies R(λ) = o(λn−1). This is a slight im-provement over (1.3).

In [We3] Weyl formulated a conjecture which claims the existence of asecond term in the asymptotic expansion for a bounded domain Ω ⊂ R3 ,namely he predicted that

N(λ) =vol(Ω)6π2 λ3 − vol(∂Ω)

16πλ2 + o(λ2).

This was proved for manifolds with boundary under a certain conditionon the periodic billiard trajectories, by Ivrii [Iv] and Melrose [Me].

The purpose of this paper is to discuss Weyl’s law in the context oflocally symmetric spaces Γ\S of finite volume and non-compact type.Here S = G/K is a Riemannian symmetric space, where G is a realsemi-simple Lie group of non-compact type, and K a maximal compactsubgroup of G. Moreover Γ is a lattice in G, i.e., a discrete subgroupof finite covolume. Of particular interest are arithmetic subgroups suchas the principal congruence subgroup Γ(N) of SL(2, Z) of level N ∈ N.Spectral theory of the Laplacian on arithmetic quotients Γ\S is inti-mately related with the theory of automorphic forms. In fact, for asymmetric space S it is more natural and important to consider notonly the Laplacian, but the whole algebra D(S) of G-invariant differ-ential operators on S. It is known that D(S) is a finitely generatedcommutative algebra [He]. Therefore, it makes sense to study the jointspectral decomposition of D(S). Square integrable joint eigenfunctionsof D(S) are examples of automorphic forms. Among them are the cuspforms which satisfy additional decay conditions. Cusps forms are thebuilding blocks of the theory of automorphic forms and, according todeep and far-reaching conjectures of Langlands [La2], are expected toprovide important relations between harmonic analysis and number the-ory.

Let G = NAK be the Iwasawa decomposition of G and let a be theLie algebra of A. If Γ\S is compact, the spectrum of D(S) in L2(Γ\S)is a discrete subset of the complexification a∗C of a∗. It has been studiedby Duistermaat, Kolk, and Varadarajan in [DKV]. The method is basedon the Selberg trace formula. The results are more refined statementsabout the distribution of the spectrum than just the Weyl law. Forexample, one gets estimations for the distribution of the tempered and

136 Werner Müller

the complementary spectrum. We will review briefly these results insection 2.

If Γ\S is non-compact, which is the case for many important arith-metic groups, the Laplacian has a large continuous spectrum which canbe described in terms of Eisenstein series [La1]. Therefore, it is not ob-vious that the Laplacian has any eigenvalue λ > 0, and an importantproblem in the theory of automorphic forms is the existence and con-struction of cusp forms for a given lattice Γ. This is were the Weyl lawcomes into play. Let H be the upper half-plane. Recall that SL(2, R) actson H by fractional linear transformations. Using his trace formula [Se2],Selberg established the following version of Weyl’s law for an arbitrarylattice Γ in SL(2, R)

NΓ(λ) + MΓ(λ) ∼ Area(Γ\H)4π

λ2 , λ →∞ (1.4)

[Se2, p. 668]. Here NΓ(λ) is the counting function of the eigenvaluesand MΓ(λ) is the winding number of the determinant ϕ(1/2+ ir) of thescattering matrix which is given by the constant Fourier coefficients ofthe Eisenstein series (see section 4). In general, the two functions on theleft can not be estimated separately. However, for congruence groups likeΓ(N), the meromorphic function ϕ(s) can be expressed in terms of well-known functions of analytic number theory. In this case, it is possible toshow that the growth of MΓ(λ) is of lower order which implies Weyl’s lawfor the counting function of the eigenvalues [Se2, p.668]. Especially itfollows that Maass cusp forms exist in abundance for congruence groups.On the other hand, there are indications [PS1], [PS2] that the existenceof many cusp forms may be restricted to arithmetic groups. This willbe discussed in detail in section 4.

In section 5 we discuss the general case of a non-compact arithmeticquotient Γ\S. There has been some recent progress with the spectralproblems discussed above. Lindenstrauss and Venkatesh [LV] estab-lished Weyl’s law without remainder term for congruence subgroups ofa split adjoint semi-simple group G. In [Mu3] this had been provedfor congruence subgroups of SL(n) and for the Bochner-Laplace op-erator acting in sections of a locally homogeneous vector bundle overSn = SL(n, R)/SO(n). For congruence subgroups of G = SL(n), an es-timation of the remainder term in Weyl’s law has been established by E.Lapid and the author in [LM]. Using the approach of [DKV] combinedwith the Arthur trace formula, the results of [DKV] have been extendedin [LM] to the cuspidal spectrum of D(Sn ).


2 Compact locally symmetric spaces

In this section we review Hörmanders method of the derivation of Weyl’slaw with remainder term for the Laplacian ∆ of a closed Riemannianmanifold M of dimension n. Then we will discuss the results of [DKV]concerning spectral asymptotics for compact locally symmetric mani-folds.

The method of Hörmander [Ho] to estimate the remainder term isbased on the study of the kernel of e−it

√∆ . The main point is the

construction of a good approximate fundamental solution to the waveequation by means of the theory of Fourier integral operators and theanalysis of the singularities of its trace

Tr e−it√

∆ =∑

j

e−it√

λj ,

which is well-defined as a distribution. The analysis of Hörmander of the“big” singularity of Tr e−it

√∆ at t = 0 leads to the following key result

[DG, (2.16)]. Let µj :=√

λj , j ∈ N. There exist cj ∈ R, j = 0, ..., n− 1,and ε > 0 such that for every h ∈ S(R) with supp h ⊂ [−ε, ε] and h ≡ 1in a neighborhood of 0 one has

∑j

h(µ− µj ) ∼ (2π)−nn−1∑k=0

ckµn−1−k , µ →∞, (2.1)

and rapidly decreasing as µ → −∞. The constants ck are of the form

ck =∫

M

ωk ,

where the ωk ’s are real valued smooth densities on M canonically asso-ciated to the Riemannian metric of M . Especially

c0 = vol(S∗M), c1 = (1− n)∫

S∗M

sub ∆,

where S∗M is the unit co-tangent bundle, and sub ∆ denotes the sub-principal symbol of ∆. Consideration of the top term in (2.1) leads tothe basic estimates for the eigenvalues.

If M = Γ\G/K is a locally symmetric manifold, the Selberg trace for-mula can be used to replace (2.1) by an exact formula [DKV]. Actually,if the rank of M is bigger than 1, the spectrum is multidimensional.Then the Selberg trace formula gives more refined information.

As example, we consider a compact hyperbolic surface M = Γ\H,

138 Werner Müller

where Γ ⊂ PSL(2, R) is a discrete, torsion-free, co-compact subgroup.Let ∆ be the hyperbolic Laplace operator which is given by

∆ = −y2(

∂2

∂x2 +∂2

∂y2

), z = x + iy. (2.2)

Write the eigenvalues λj of ∆ as

λj =14

+ r2j ,

where rj ∈ C and arg(rj ) ∈ 0, π/2. Let h be an analytic function in astrip | Im(z)| ≤ 1

2 + δ, δ > 0, such that

h(z) = h(−z), |h(z)| ≤ C(1 + |z|)−2−δ . (2.3)

Let

g(u) =12π

∫R

h(r)eirudr.

Given γ ∈ Γ denote by γΓ its Γ-conjugacy class. Since Γ is co-compact,each γ = e is hyperbolic. Each hyperbolic element γ is the power of aprimitive hyperbolic element γ0 . A hyperbolic conjugacy class deter-mines a closed geodesic τγ of Γ\H. Let l(γ) denote the length of τγ .Then the Selberg trace formula [Se1] is the following identity:

∞∑j=0

h(rj )

=Area(Γ\H)

4π

∫R

h(r)r tanh(πr) dr +∑

γΓ =e

l(γ0)

2 sinh(

l(γ )2

)g(l(γ)).

(2.4)

Now let g ∈ C∞c (R) and h(z) =

∫R

g(r)e−irz dr. Then h is entire andrapidly decreasing in each strip | Im(z)| ≤ c, c > 0. Let t ∈ R and set

ht(z) = h(t− z) + h(t + z).

Then ht is entire and satisfies (2.3). Note that ht(r) = e−itr g(r) +eitr g(−r). We symmetrize the spectrum by r−j := −rj , j ∈ N. Then by


(2.4) we get∞∑

j=−∞h(t− rj )

=Area(Γ\H)

2π

∫R

h(t− r)r tanh(πr) dr

+∑

γΓ =e

l(γ0)

2 sinh(

l(γ )2

) (e−itl(γ )g(l(γ)) + eitl(γ )g(−l(γ))

).

(2.5)

Let ε > 0 be such that l(γ) > ε for all hyperbolic conjugacy classesγΓ . The following lemma is an immediate consequence of (2.5).

Lemma 2.1 Let g ∈ C∞c (R) such that supp g ⊂ (−ε, ε). Let h(z) =∫

Rg(r)e−irz dr. Then for all t ∈ R we have

∞∑j=−∞

h(t− rj ) =Area(Γ\H)

2π

∫R

h(t− r)r tanh(πr) dr. (2.6)

Changing variables in the integral on the right and using that

tanh(π(r + t)) = 1− 2e−2π (r+t)

1 + e−2π (r+t) = −1 +2e2π (r+t)

1 + e2π (r+t) ,

we obtain the following asymptotic expansion∞∑

j=−∞h(t− rj )

=Area(Γ\H)

2π

(|t|∫

R

h(r) dr − sign t

∫R

h(r)r dr

)+ O

(e−2π |t|

),

(2.7)

as |t| → ∞. If h is even, the second term vanishes and the asymptoticexpansion is related to (2.1). The asymptotic expansion (2.7) can beused to derive estimates for the number of eigenvalues near a given pointµ ∈ R.

Lemma 2.2 For every a > 0 there exists C > 0 such that

#j : |rj − µ| ≤ a ≤ C(1 + |µ|)

for all µ ∈ R.

Proof We proceed as in the proof of Lemma 2.3 in [DG]. As shown

140 Werner Müller

in the proof, there exists h ∈ S(R) such that h ≥ 0, h > 0 on [−a, a],h(0) = 1, and supp h is contained in any prescribed neighborhood of 0.Now observe that there are only finitely many eigenvalues λj = 1/4+ r2

j

with rj /∈ R. Therefore it suffices to consider rj ∈ R. Let µ ∈ R. By(2.7) we get

#j : |rj − µ| ≤ a, rj ∈ R ·minh(u) : |u| ≤ a≤

∑rj ∈R

h(µ− rj ) ≤ C(1 + |µ|).

This lemma is the basis of the following auxiliary results.

Lemma 2.3 For every h as above there exists C > 0 such that∑|rj |≤λ

∣∣∣∣ ∫R−[−λ,λ ]

h(t− rj ) dt

∣∣∣∣ ≤ Cλ,∑

|rj |>λ

∣∣∣∣ ∫ λ

−λ

h(t− rj ) dt

∣∣∣∣ ≤ Cλ,

(2.8)for all λ ≥ 1.

Proof Since h is rapidly decreasing, there exists C > 0 such that |h(t)| ≤C(1 + |t|)−4 , t ∈ R. Let [λ] be the largest integer ≤ λ. Then we get∑|rj |≤λ

∣∣∣∣ ∫ ∞

λ

h(t− rj ) dt

∣∣∣∣ ≤ ∑|rj |≤λ

∫ ∞

λ−rj

|h(t)| dt ≤ C∑

|rj |≤λ

1(1 + λ− rj )3

=[λ ]−1∑

k=−[λ ]

∑k≤rj ≤k+1

1(1 + λ− rj )3 ≤

[λ ]−1∑k=−[λ ]

#j : |rj − k| ≤ 1(λ− k)3 ,

and by Lemma 2.2 the right hand side is bounded by Cλ for λ ≥ 1.Similarly we get ∑

|rj |≤λ

∣∣∣∣ ∫ −λ

−∞h(t− rj ) dt

∣∣∣∣ ≤ C2λ.

The second series can be treated in the same way.

Lemma 2.4 Let h be as in Lemma 2.1 and such that h(0) = 1. Then∫ λ

−λ

∞∑j=−∞

h(t− rj ) dt =Area(Γ\H)

2πλ2 + O(λ) (2.9)

as λ →∞.


Proof To prove the lemma, we integrate (2.6) and determine the asymp-totic behavior of the integral on the right. Let p(r) be a continuousfunction on R such that |p(r)| ≤ C(1+ |r|) and p(r) = p(−r). Changingthe order of integration and using that

∫R

h(t− r) dt = h(0) = 1, we get∫ λ

−λ

∫R

h(t− r)p(r) dr dt =∫ λ

−λ

p(r) dr

−∫ λ

−λ

(∫R−[−λ,λ ]

h(t− r) dt

)p(r) dr

+∫

R−[−λ,λ ]

(∫ λ

−λ

h(t− r) dt

)p(r) dr.

Let C1 > 0 be such that |h(r)| ≤ C1(1 + |r|)−3 , r ∈ R. Then the secondand the third integral can be estimated by C(1 + λ). Thus we get∫ λ

−λ

∫R

h(t− r)p(r) dr dt =∫ λ

−λ

p(r) dr + O(λ), λ →∞. (2.10)

If we apply (2.10) to p(r) = r tanh(πr), we obtain∫ λ

−λ

∫R

h(t− r)r tanh(πr) dr dt = λ2 + O(λ). (2.11)

This proves the lemma.

We are now ready to prove Weyl’s law. We choose h such that h hassufficiently small support and h(0) = 1. Then∫ λ

−λ

∞∑j=−∞

h(t− rj ) dt =∑

|rj |≤λ

∫R

h(t− rj ) dt

−∑

|rj |≤λ

∫R−[−λ,λ ]

h(t− rj ) dt +∑

|rj |>λ

∫ λ

−λ

h(t− rj ) dt.

Using that∫

Rh(t− r) dt = h(0) = 1, we get

2NΓ(λ) =∫ λ

−λ

∑j

h(t− rj ) dt +∑

|rj |≤λ

∫R−[−λ,λ ]

h(t− rj ) dt

−∑

|rj |>λ

∫ λ

−λ

h(t− rj ) dt.

142 Werner Müller

By Lemmas 2.3 and 2.4 we obtain

NΓ(λ) =Area(Γ\H)

4πλ2 + O(λ). (2.12)

We turn now to an arbitrary Riemannian symmetric space S = G/K

of non-compact type and we review the main results of [DKV]. Thegroup of motions G of S is a semi-simple Lie group of non-compact typewith finite center and K is a maximal compact subgroup of G. TheLaplacian ∆ of S is a G-invariant differential operator on S, i.e., ∆commutes with the left translations Lg , g ∈ G. Besides of ∆ we needto consider the ring D(S) of all invariant differential operators on S.It is well-known that D(S) is commutative and finitely generated. Itsstructure can be described as follows. Let G = NAK be the Iwasawadecomposition of G, W the Weyl group of (G,A) and a be the Liealgebra of A. Let S(aC) be the symmetric algebra of the complexificationaC = a⊗C of a and let S(aC)W be the subspace of Weyl group invariantsin S(aC). Then by a theorem of Harish-Chandra [He, Ch. X, Theorem6.15] there is a canonical isomorphism

µ : D(S) ∼= S(aC)W . (2.13)

This shows that D(S) is commutative. The minimal number of genera-tors equals the rank of S which is dim a [He, Ch.X, §6.3]. Let λ ∈ a∗C.Then by (2.13), λ determines an character

χλ : D(S)→ C

and χλ = χλ ′ if and only if λ and λ′ are in the same W -orbit. SinceS(aC) is integral over S(aC)W [He, Ch. X, Lemma 6.9], each characterof D(S) is of the form χλ for some λ ∈ a∗C. Thus the characters of D(S)are parametrized by a∗C/W .

Let Γ ⊂ G be a discrete, torsion-free, co-compact subgroup of G.Then Γ acts properly discontinuously on S without fixed points and thequotient M = Γ\S is a locally symmetric manifold which is equippedwith the metric induced from the invariant metric of S. Then eachD ∈ D(S) descends to a differential operator

D : C∞(Γ\S) → C∞(Γ\S).

Let E ⊂ C∞(Γ\S) be an eigenspace of the Laplace operator. Then Eis a finite-dimensional vector space which is invariant under D ∈ D(S).For each D ∈ D(S), the formal adjoint D∗ of D also belongs to D(S).


Thus we get a representation

ρ : D(S)→ End(E)

by commuting normal operators. Therefore, E decomposes into the di-rect sum of joint eigenspaces of D(S). Given λ ∈ a∗C/W , let

E(λ) = ϕ ∈ C∞(Γ\S) : Dϕ = χλ(D)ϕ, D ∈ D(S).

Let m(λ) = dim E(λ). Then the spectrum Λ(Γ) of Γ\S is defined to be

Λ(Γ) = λ ∈ a∗C/W : m(λ) > 0,

and we get an orthogonal direct sum decomposition

L2(Γ\S) =⊕

λ∈Λ(Γ)

E(λ).

If we pick a fundamental domain for W , we may regard Λ(Γ) as asubset of a∗C. If rk (S) > 1, then Λ(Γ) is multidimensional. Again thedistribution of Λ(Γ) is studied using the Selberg trace formula [Se1]. Todescribe it we need to introduce some notation. Let C∞

c (G//K) be thesubspace of all f ∈ C∞

c (G) which are K-bi-invariant. Let

A : C∞c (G//K) → C∞

c (A)W

be the Abel transform which is defined by

A(f)(a) = δ(a)1/2∫

N

f(an) dn, a ∈ A,

where δ is the modulus function of the minimal parabolic subgroup P =NA. Given h ∈ C∞

c (A)W , let

h(λ) =∫

A

h(a)e〈λ,H (a)〉 da.

Let β(iλ), λ ∈ a∗, be the Plancherel density. Then the Selberg traceformula is the following identity∑

λ∈Λ(Γ)

m(λ)h(λ) =vol(Γ\G)|W |

∫a∗

h(λ)β(iλ) dλ

+∑

[γ ]Γ =e

vol(Γγ \Gγ )∫

Gγ \G

A−1(h)(x−1γx) dγ x.

(2.14)

144 Werner Müller

This is still not the final form of the Selberg trace formula. The distri-butions

Jγ (f) = vol(Γγ \Gγ )∫

Gγ \G

f(x−1γx) dγ x, f ∈ C∞c (G), (2.15)

are invariant distribution an G and can be computed using Harish-Chandra’s Fourier inversion formula. This brings (2.14) into a formwhich is similar to (2.4). For the present purpose, however, it suffices towork with (2.14). Since for γ = e, the conjugacy class of γ in G is closedand does not intersect K, there exists an open neighborhood V of 1 inA satisfying V = V −1 , V is invariant under W , and Jγ (A−1(h)) = 0 forall h ∈ C∞

c (V ) [DKV, Propostion 3.8]. Thus we get∑λ∈Λ(Γ)

m(λ)h(λ) =vol(Γ\G)|W |

∫a∗

h(λ)β(iλ) dλ (2.16)

for all h ∈ C∞c (V ). One can now proceed as in the case of the upper

half-plane. The basic step is again to estimate the number of λ ∈ Λ(Γ)lying in a ball of radius r around a variable point µ ∈ ia∗. This canbe achieved by inserting appropriate test functions h into (2.16) [DKV,section 7]. Let

Λtemp(Γ) = Λ(Γ) ∩ ia∗, Λcomp(Γ) = Λ(Γ) \ Λtemp(Γ)

be the tempered and complementary spectrum, respectively. Given anopen bounded subset Ω of a∗ and t > 0, let

tΩ := tµ : µ ∈ Ω. (2.17)

One of the main results of [DKV] is the following asymptotic formulafor the distribution of the tempered spectrum [DKV, Theorem 8.8]∑

λ∈Λtemp(Γ)∩(itΩ)

m(λ) =vol(Γ\G)|W |

∫itΩ

β(iλ) dλ + O(tn−1), t→∞,

(2.18)Note that the leading term is of order O(tn ). The growth of the com-plementary spectrum is of lower order. Let Bt(0) ⊂ a∗C be the ball ofradius t > 0 around 0. There exists C > 0 such that for all t ≥ 1∑

λ∈Λcomp(Γ)∩Bt (0)

m(λ) ≤ Ctn−2 (2.19)

[DKV, Theorem 8.3]. The estimates (2.18) and (2.19) contain more in-formation about the distribution of Λ(Γ) than just the Weyl law. Indeed,the eigenvalue of ∆ corresponding to λ ∈ Λtemp(Γ) equals ‖λ‖2 +‖ρ‖2 .


So if we choose Ω in (2.18) to be the unit ball, then (2.18) together with(2.19) reduces to Weyl’s law for Γ\S.

We note that (2.18) and (2.19) can also be rephrased in terms ofrepresentation theory. Let R be the right regular representation of G inL2(Γ\G) defined by

(R(g1)f)(g2) = f(g2g1), f ∈ L2(Γ\G), g1 , g2 ∈ G.

Let G be the unitary dual of G, i.e., the set of equivalence classes ofirreducible unitary representations of G. Since Γ\G is compact, it iswell known that R decomposes into direct sum of irreducible unitaryrepresentations of G. Given π ∈ G, let m(π) be the multiplicity withwhich π occurs in R. Let Hπ denote the Hilbert space in which π acts.Then

L2(Γ\G) ∼=⊕π∈G

m(λ)Hπ .

Now observe that L2(Γ\S) = L2(Γ\G)K . Let HKπ denote the subspace

of K-fixed vectors in Hπ . Then

L2(Γ\S) ∼=⊕π∈G

m(λ)HKπ .

Note that dimHKπ ≤ 1. Let G(1) ⊂ G be the subset of all π with

HKπ = 0. This is the spherical dual. Given π ∈ G, let λπ be the

infinitesimal character of π. If π ∈ G(1), then λπ ∈ a∗C/W . Moreoverπ ∈ G(1) is tempered if π is unitarily induced from the minimal parabolicsubgroup P = NA. In this case we have λπ ∈ ia∗/W . So (2.18) can berewritten as∑

π∈G(1)λπ ∈itΩ

m(π) =vol(Γ\G)|W |

∫itΩ

β(λ) dλ + O(tn−1), t→∞. (2.20)

3 Automorphic forms

The theory of automorphic forms is concerned with harmonic analysison locally symmetric spaces Γ\S of finite volume. Of particular interestare arithmetic groups Γ. This means that we consider a connected semi-simple algebraic group G defined over Q such that G = G(R) and Γ isa subgroup of G(Q) which is commensurable with G(Z), where G(Z)is defined with respect to some embedding G ⊂ GL(m). The standardexample is G = SL(n) and Γ(N) ⊂ SL(n, Z) the principal congruence

146 Werner Müller

subgroup of level N . A basic feature of arithmetic groups is that thequotient Γ\S has finite volume [BH]. Moreover in many important casesit is non-compact. A typical example for that is Γ(N)\SL(n, R)/SO(n).

In this section we discuss only the case of the upper half-plane H andwe consider congruence subgroups of SL(2, Z). For N ≥ 1 the principalcongruence subgroup Γ(N) of level N is defined as

Γ(N) =γ ∈ SL(2, Z) : γ ≡ Id mod N

.

A congruence subgroup Γ of SL(2, Z) is a subgroup for which there existsN ∈ N such that Γ contains Γ(N). An example of a congruence subgroupis the Hecke group

Γ0(N) =(

a b

c d

)∈ SL(2, Z) : c ≡ 0 mod N

.

If Γ is torsion free, the quotient Γ\H is a finite area, non-compact,hyperbolic surface. It has a decomposition

Γ\H = M0 ∪ Y1 ∪ · · · ∪ Ym , (3.1)

into the union of a compact surface with boundary M0 and a finite num-ber of ends Yi

∼= [a,∞)×S1 which are equipped with the Poincaré metric.In general, Γ\H may have a finite number of quotient singularities. Thequotient Γ(N)\H is the modular surface X(N).

Let ∆ be the hyperbolic Laplacian (2.2). A Maass automorphic formis a smooth function f : H → C which satisfies

a) f(γz) = f(z), γ ∈ Γ.b) There exists λ ∈ C such that ∆f = λf .c) f is slowly increasing.

Here the last condition means that there exist C > 0 and N ∈ N suchthat the restriction fi of f to Yi satisfies

|fi(y, x)| ≤ CyN , y ≥ a, i = 1, ...,m.

Examples are the Eisenstein series. Let a1 , ..., am ∈ R ∪ ∞ be repre-sentatives of the Γ-conjugacy classes of parabolic fixed points of Γ. Theai ’s are called cusps. For each ai let Γai

be the stabilizer of ai in Γ.Choose σi ∈ SL(2, R) such that

σi(∞) = ai, σ−1i Γai

σi =(

1 n

0 1

): n ∈ Z

.


Then the Eisenstein series Ei(z, s) associated to the cusp ai is definedas

Ei(z, s) =∑

γ∈Γa i\Γ

Im(σ−1i γz)s , Re(s) > 1. (3.2)

The series converges absolutely and uniformly on compact subsets of thehalf-plane Re(s) > 1 and it satisfies the following properties.

1) Ei(γz, s) = Ei(z, s) for all γ ∈ Γ.2) As a function of s, Ei(z, s) admits a meromorphic continuation to

C which is regular on the line Re(s) = 1/2.3) Ei(z, s) is a smooth function of z and satisfies

∆zEi(z, s) = s(1− s)Ei(z, s).

As an example consider the modular group Γ(1) which has a single cusp∞. The Eisenstein series attached to ∞ is the well-known series

E(z, s) =∑

(m,n)∈Z2

(m,n)=1

ys

|mz + n|2s.

In the general case, the Eisenstein series were first studied by Selberg[Se1]. The Eisenstein series are closely related with the study of thespectral resolution of ∆. Regarded as unbounded operator

∆: C∞c (Γ\H) → L2(Γ\H),

∆ is essentially self-adjoint [Roe]. Let ∆ be the unique self-adjoint ex-tension of ∆. The important new feature due to the non-compactnessof Γ\H is that ∆ has a large continuous spectrum which is governed bythe Eisenstein series. The following basic result is due to Roelcke [Roe].

Proposition 3.1 The spectrum of ∆ is the union of a pure point spec-trum σpp(∆) and an absolutely continuous spectrum σac(∆).1) The pure point spectrum consists of eigenvalues 0 = λ0 < λ1 ≤ · · · offinite multiplicities with no finite points of accumulation.2) The absolutely continuous spectrum equals [1/4,∞) with multiplicityequal to the number of cusps of Γ\H.

Of particular interest are the eigenfunctions of ∆. They are Maass auto-morphic forms. This can be seen by studying the Fourier expansion of an

148 Werner Müller

eigenfunction in the cusps. As an example consider f ∈ C∞(Γ0(N)\H)which satisfies

∆f = λf, f(z) = f(−z),∫

Γ0 (N )\H

|f(z)|2 dA(z) <∞.

Assume that λ = 1/4 + r2 , r ∈ R. Then f(x + iy) admits a Fourierexpansion w.r.t. x of the form

f(x + iy) =∞∑

n=1

a(n)√

yKir (2πny) cos(2πnx), (3.3)

where Kν (y) is the modified Bessel function which may be defined by

Kν (y) =∫ ∞

0e−y cosh t cosh(νt) dt

and it satisfies

K ′′ν (y) +

1yK ′

ν (y) +(

1− ν2

y2

)Kν (y) = 0.

Now note that Kν (y) = O(e−cy ) as y → ∞. This implies that f israpidly decreasing in the cusp ∞. A similar Fourier expansion holds inthe other cusps. This implies that f is rapidly decreasing in all cuspsand therefore, it is a Maass automorphic form. In fact, since the zeroFourier coefficients vanish in all cusps, f is a Maass cusp form. Ingeneral, the space of cusp forms L2

cus(Γ\H) is defined as the subspace ofall f ∈ L2(Γ\H) such that for almost all y ∈ R+ :∫ 1

0f(σk (x + iy)) dx = 0, k = 1, ...,m.

This is an invariant subspace of ∆ and the restriction of ∆ to L2cus(Γ\H)

has pure point spectrum, i.e., L2cus(Γ\H) is the span of square integrable

eigenfunctions of ∆. Let L2res(Γ\H) be the orthogonal complement of

L2cus(Γ\H) in L2(Γ\H). Thus

L2pp(Γ\H) = L2

cus(Γ\H)⊕ L2res(Γ\H).

The subspace L2res(Γ\H) can be described as follows. The poles of the

Eisenstein series Ei(z, s) in the half-plane Re(s) > 1/2 are all simpleand are contained in the interval (1/2, 1]. Let s0 ∈ (1/2, 1] be a pole ofEi(z, s) and put

ψ = Ress=s0 Ei(z, s).

Then ψ is a square integrable eigenfunction of ∆ with eigenvalue λ =


s0(1− s0). The set of all such residues of the Eisenstein series Ei(z, s),i = 1, ...,m, spans L2

res(Γ\H). This is a finite-dimensional space whichis called the residual subspace. The corresponding eigenvalues form theresidual spectrum of ∆. So we are left with the cuspidal eigenfunctionsor Maass cusp forms. Cusp forms are the building blocks of the theoryof automorphic forms. They play an important role in number theory.To illustrate this consider an even Maass cusp form f for Γ(1) witheigenvalue λ = 1/4 + r2 , r ∈ R. Let a(n), n ∈ N, be the Fouriercoefficients of f given by (3.3). Put

L(s, f) =∞∑

n=1

a(n)ns

, Re(s) > 1.

This Dirichlet series converges absolutely and uniformly in the half-planeRe(s) > 1. Let

Λ(s, f) = π−sΓ(

s + ir

2

)Γ(

s− ir

2

)L(s, f). (3.4)

Then the modularity of f implies that Λ(s, f) has an analytic continu-ation to the whole complex plane and satisfies the functional equation

Λ(s, f) = Λ(1− s, f)

[Bu, Proposition 1.9.1]. Under additional assumptions on f , the Dirich-let series L(s, f) is also an Euler product. This is related to the arith-metic nature of the groups Γ(N). The surfaces X(N) carry a family ofalgebraically defined operators Tn , the so called Hecke operators, whichfor (n,N) = 1 are defined by

Tnf(z) =1√n

∑ad=n

b mod d

f

(az + b

d

).

These are closely related to the cosets of the finite index subgroups(n 00 1

)Γ(N)

(n 00 1

)−1

∩ Γ(N)

of Γ(N). Each Tn defines a linear transformation of L2(X(N)). The Tn ,n ∈ N, are a commuting family of normal operators which also commutewith ∆. Therefore, each Tn leaves the eigenspaces of ∆ invariant. So wemay assume that f is a common eigenfunction of ∆ and Tn , n ∈ N:

∆f = (1/4 + r2)f, Tnf = λ(n)f.

150 Werner Müller

If f = 0, then a(1) = 0. So we can normalize f such that a(1) = 1.Then it follows that a(n) = λ(n) and the Fourier coefficients satisfy thefollowing multiplicative relations

a(m)a(n) =∑

d|(m,n)

a(mn

d2

).

This implies that L(s, f) is an Euler product

L(s, f) =∞∑

n=1

a(n)n−s =∏p

(1− a(p)p−s + p−2s

)−1, (3.5)

which converges for Re(s) > 1. L(s, f) is the basic example of an au-tomorphic L-function. It is convenient to write this Euler product in adifferent way. Introduce roots αp, βp by

αpβp = 1, αp + βp = a(p).

Let

Ap =(

αp 00 βp

).

Then

L(s, f) =∏p

det(Id−App

−s)−1

.

Now let ρ : GL(2, C) → GL(N, C) be a representation. Then we canform a new Euler product by

L(s, f, ρ) =∏p

det(Id−ρ(Ap)p−s

)−1,

which converges in some half-plane. It is part of the general conjec-tures of Langlands [La2] that each of these Euler products admits ameromorphic extension to C and satisfies a functional equation. Theconstruction of Euler products for Maass cusp forms can be extended toother groups G, in particular to cusp forms on GL(n). It is also con-jectured that L(s, f, ρ) is an automorphic L-function of an automorphicform on some GL(n). This is part of the functoriality principle of Lang-lands. Furthermore, all L-functions that occur in number theory andalgebraic geometry are expected to be automorphic L-functions. Thisis one of the main reasons for the interest in the study of cusp forms.Other applications are discussed in [Sa1].


4 The Weyl law and existence of cusp forms

Since Γ(N)\H is not compact, it is not clear that there exist any eigen-values λ > 0. By Proposition 3.1 the continuous spectrum of ∆ equals[1/4,∞). Thus all eigenvalues λ ≥ 1/4 are embedded in the continu-ous spectrum. It is well-known in mathematical physics, that embeddedeigenvalues are unstable under perturbations and therefore, are difficultto study.

One of the basic tools to study the cuspidal spectrum is the Selbergtrace formula [Se2]. The new terms in the trace formula, which are due tothe non-compactness of Γ\H arise from the parabolic conjugacy classesin Γ and the Eisenstein series. The contribution of the Eisenstein seriesis given by their zeroth Fourier coefficients of the Fourier expansion inthe cusps. The zeroth Fourier coefficient of the Eisenstein series Ek (z, s)in the cusp al is given by∫ 1

0Ek (σl(x + iy), s) dx = ys + Ckl(s)y1−s ,

where Ckl(s) is a meromorphic function of s ∈ C. Put

C(s) := (Ckl(s))mk,l=1 .

This is the so called scattering matrix. Let

ϕ(s) := detC(s).

Let the notation be as in (2.4) and assume that Γ has no torsion. Thenthe trace formula is the following identity.∑

j

h(rj ) =Area(Γ\H)

4π

∫R

h(r)r tanh(πr) dr +∑γΓ

l(γ0)

2 sinh(

l(γ )2

)g(l(γ))

+14π

∫ ∞

−∞h(r)

ϕ′

ϕ(1/2 + ir) dr − 1

4ϕ(1/2)h(0)

− m

2π

∫ ∞

−∞h(r)

Γ′

Γ(1 + ir)dr +

m

4h(0)−m ln 2 g(0).

(4.1)

The trace formula holds for every discrete subgroup Γ ⊂ SL(2, R) withfinite coarea. In analogy to the counting function of the eigenvalues weintroduce the winding number

MΓ(λ) = − 14π

∫ λ

−λ

ϕ′

ϕ(1/2 + ir) dr

152 Werner Müller

which measures the continuous spectrum. Using the cut-off Laplacianof Lax-Phillips [CV] one can deduce the following elementary bounds

NΓ(λ) ! λ2 , MΓ(λ) ! λ2 , λ ≥ 1. (4.2)

These bounds imply that the trace formula (4.1) holds for a larger classof functions. In particular, it can be applied to the heat kernel kt(u). Itsspherical Fourier transform equals ht(r) = e−t(1/4+r 2 ) , t > 0. If we insertht into the trace formula we get the following asymptotic expansion ast → 0.∑

j

e−tλj − 14π

∫R

e−t(1/4+r 2 ) ϕ′

ϕ(1/2 + ir) dr

=Area(Γ\H)

4πt+

a log t√t

+b√t

+ O(1)

(4.3)

for certain constants a, b ∈ R. Using [Se2, (8.8), (8.9)] it follows that thewinding number MΓ(λ) is monotonic increasing for r " 0. Therefore wecan apply a Tauberian theorem to (4.3) and we get the Weyl law (1.4).

In general, we cannot estimate separately the counting function andthe winding number. For congruence subgroups, however, the entries ofthe scattering matrix can be expressed in terms of well-known analyticfunctions. For Γ(N) the determinant of the scattering matrix ϕ(s) hasbeen computed by Huxley [Hu]. It has the form

ϕ(s) = (−1)lA1−2s

(Γ(1− s)

Γ(s)

)k ∏χ

L(2− 2s, χ)L(2s, χ)

, (4.4)

where k, l ∈ Z, A > 0, the product runs over Dirichlet characters χ tosome modulus dividing N and L(s, χ) is the Dirichlet L-function withcharacter χ. Especially for Γ(1) we have

ϕ(s) =√

πΓ(s− 1/2)ζ(2s− 1)

Γ(s)ζ(2s), (4.5)

where ζ(s) denotes the Riemann zeta function.Using Stirling’s approximation formula to estimate the logarithmic

derivative of the Gamma function and standard estimations for the log-arithmic derivative of Dirichlet L-functions on the line Re(s) = 1 [Pr,Theormem 7.1], we get

ϕ′

ϕ(1/2 + ir) = O(log(4 + |r|)), |r| → ∞. (4.6)


This implies that

MΓ(N )(λ) ! λ log λ. (4.7)

Together with (1.4) we obtain Weyl’s law for the point spectrum

NΓ(N )(λ) ∼ Area(X(N))4π

λ2 , λ →∞, (4.8)

which is due to Selberg [Se2, p.668]. A similar formula holds for othercongruence groups such as Γ0(N). In particular, (4.8) implies that forcongruence groups Γ there exist infinitely many linearly independentMaass cusp forms.

A proof of the Weyl law (4.8) which avoids the use of the constantterms of the Eisenstein series has recently been given by Lindenstraussand Venkatesh [LV]. Their approach is based on the construction ofconvolution operators with purely cuspidal image.

Neither of these methods give good estimates of the remainder term.One approach to obtain estimates of the remainder term is based on theSelberg zeta function

ZΓ(s) =∏γΓ

∞∏k=0

(1− e−(s+k)(γ )

), Re(s) > 1,

where the outer product runs over the primitive hyperbolic conjugacyclasses in Γ and (γ) is the length of the closed geodesic associated toγΓ . The infinite product converges absolutely in the indicated half-plane and admits an analytic continuation to the whole complex plane. Ifλ = 1/4 + r2 , r ∈ R∪ i(1/2, 1], is an eigenvalue of ∆, then s0 = 1/2 + ir

is a zero of ZΓ(s). Using this fact and standard methods of analyticnumber theory one can derive the following strong form of the Weyl law[Hj, Theorem 2.28], [Ve, Theorem 7.3].

Theorem 4.1 Let m be the number of cusps of Γ\H. There exists c > 0such that

NΓ(λ) + MΓ(λ) =Area(Γ\H)

4πλ2 − m

πλ log λ + cλ + O

(λ(log λ)−1)

as λ →∞.

Together with (4.7) we obtain Weyl’s law with remainder term.

Theorem 4.2 For every N ∈ N we have

NΓ(N )(λ) =Area(X(N))

4πλ2 + O(λ log λ) as λ →∞.

154 Werner Müller

The use of the Selberg zeta function to estimate the remainder termis limited to rank one cases. However, the remainder term can also beestimated by Hörmander’s method using the trace formula as in thecompact case. We describe the main steps. Let ε > 0 such that (γ) > ε

for all hyperbolic conjugacy classes γΓ(N ) . Choose g ∈ C∞c (R) to be

even and such that supp g ⊂ (−ε, ε). Let h(z) =∫

Rg(r)e−irz dr. Then

the hyperbolic contribution in the trace formula (4.1) drops out. Wesymmetrize the spectrum by r−j = −rj , j ∈ N. Then for each t ∈ R wehave∑

j

h(t− rj ) =Area(X(N))

2π

∫R

h(t− r)r tanh(πr) dr

+12π

∫ ∞

−∞h(t− r)

ϕ′

ϕ(1/2 + ir) dr − 1

2ϕ(1/2)h(t)

− m

π

∫ ∞

−∞h(t− r)

Γ′

Γ(1 + ir)dr +

m

2h(t)− 2m ln 2 g(0).

(4.9)

Now we need to estimate the behavior of the terms on the right handside as |t| → ∞. The first integral has been already considered in (2.7).It is of order O(|t|). To deal with the second integral we use (4.6). Thisimplies∫

R

h(t− r)ϕ′

ϕ(1/2 + ir) dr = O(log(|t|)), |t| → ∞. (4.10)

Using Stirling’s formula we get∫R

h(t− r)Γ′

Γ(1 + ir) dr = O(log(|t|)), |t| → ∞.

The remaining terms are bounded. Combining these estimations, we get∑j

h(t− rj ) = O(|t|), |t| → ∞.

Therefore, Lemma 2.2 holds also in the present case. It remains toestablish the analog of Lemma 2.4. Using (2.10) and (4.6) we get∫ λ

−λ

∫R

h(t− r)ϕ′

ϕ(1/2 + it) dr dt = O(λ log λ). (4.11)

Similarly, using Stirling’s formula and (2.10), we obtain∫ λ

−λ

∫R

h(t− r)Γ′

Γ(1 + it) dr dt = O(λ log λ). (4.12)


The integral of the remaining terms is of order O(λ). Thus we obtain∫ λ

−λ

∞∑j=−∞

h(t− rj ) dt =Area(X(N))

2πλ2 + O(λ log λ) (4.13)

as λ →∞. Now we proceed in exactly the same way as in the compactcase. Using Lemma 2.3 and (4.13), Theorem 4.2 follows.

The Weyl law shows that for congruence groups Maass cusp formsexist in abundance. In general very little is known. Let Γ be any discrete,co-finite subgroup of SL(2, R). Then by Donnelly [Do] the followinggeneral bound is known

lim supλ→∞

N cusΓ (λ)λ

≤ Area(Γ\H)4π

.

A group Γ for which the equality is attained is called essentially cuspidalby Sarnak [Sa2]. By (4.8), Γ(N) is essentially cuspidal. The study of thebehavior of eigenvalues under deformations of Γ, initiated Phillips andSarnak [PS1], [PS2], supports the conjecture that essential cuspidalitymay be limited to special arithmetic groups.

The consideration of the behavior of cuspidal eigenvalues under de-formations was started by Colin de Verdiere [CV] in the more generalcontext of metric perturbations. One of his main results [CV, Théorème7] states that under a generic compactly supported conformal pertur-bation of the hyperbolic metric of Γ\H all Maass cusp forms are dis-solved. This means that each point sj = 1/2 + irj , rj ∈ R, such thatλj = sj (1 − sj ) is an eigenvalue moves under the perturbation into thehalf-plane Re(s) < 1/2 and becomes a pole of the scattering matrixC(s).

In the present context we are only interested in deformations such thatthe curvature stays constant. Such deformations are given by curves inthe Teichmüller space T (Γ) of Γ. The space T (Γ) is known to be afinite-dimensional and therefore, it is by no means clear that the resultsof [CV] will continue to hold for perturbations of this restricted type.For Γ(N) this problem has been studied in [PS1], [PS2]. One of themain results is an analog of Fermi’s golden rule which gives a sufficientcondition for a cusp form of Γ(N) to be dissolved under a deformation inT (Γ(N)). Based on these results, Sarnak made the following conjecture[Sa2]:

156 Werner Müller

Conjecture.

(a) The generic Γ in a given Teichmüller space of finite area hyperbolicsurfaces is not essentially cuspidal.

(b) Except for the Teichmüller space of the once punctured torus, thegeneric Γ has only finitely many eigenvalues.

5 Higher rank

In this section we consider an arbitrary locally symmetric space Γ\Sdefined by an arithmetic subgroup Γ ⊂ G(Q), where G is a semi-simplealgebraic group over Q with finite center, G = G(R) and S = G/K.The basic example will be G = SL(n) and Γ = Γ(N), the principalcongruence subgroup of SL(n, Z) of level N which consists of all γ ∈SL(n, Z) such that γ ≡ Id mod N .

Let ∆ be the Laplacian of Γ\S, and let ∆ be the closure of ∆ in L2 .Then ∆ is a non-negative self-adjoint operator in L2(Γ\S). The proper-ties of its spectral resolution can be derived from the known structure ofthe spectral resolution of the regular representation RΓ of G on L2(Γ\G)[La1], [BG]. In this way we get the following generalization of Proposi-tion 3.1.

Proposition 5.1 The spectrum of ∆ is the union of a point spectrumσpp(∆) and an absolutely continuous spectrum σac(∆).1) The point spectrum consists of eigenvalues 0 = λ0 < λ1 ≤ · · · of finitemultiplicities with no finite point of accumulation.2) The absolutely continuous spectrum equals [b,∞) for some b > 0.

The theory of Eisenstein series [La1] provides a complete set of gener-alized eigenfunctions for ∆. The corresponding wave packets span theabsolutely continuous subspace L2

c (Γ\S). This allows us to determinethe constant b explicitly in terms of the root structure. The statementabout the point spectrum was proved in [BG, Theorem 5.5].

Let L2dis(Γ\S) be the closure of the span of all eigenfunctions. It

contains the subspace of cusp forms L2cus(Γ\S). We recall its definition.

Let P ⊂ G be a parabolic subgroup defined over Q [Bo]. Let P = P(R).This is a cuspidal parabolic subgroup of G and all cuspidal parabolicsubgroups arise in this way. Let NP be the unipotent radical of P andlet NP = NP (R). Then NP ∩ Γ\NP is compact. A cusp form is asmooth function ϕ on Γ\S which is a joint eigenfunction of the ring


D(S) of invariant differential operators on S, and which satisfies∫NP ∩Γ\NP

ϕ(nx) dn = 0 (5.1)

for all cuspidal parabolic subgroups P = G. Each cusp form is rapidlydecreasing and hence square integrable. Let L2

cus(Γ\S) be the closurein L2(Γ\S) of the linear span of all cusp forms. Then L2

cus(Γ\S) is aninvariant subspace of ∆ which is contained in L2

dis(Γ\S).Let L2

res(Γ\S) be the orthogonal complement of L2cus(Γ\S) in L2

dis(Γ\S),i.e., we have an orthogonal decomposition

L2dis(Γ\S) = L2

cus(Γ\S)⊕ L2res(Γ\S).

It follows from Langlands’s theory of Eisenstein systems that L2res(Γ\S)

is spanned by iterated residues of cuspidal Eisenstein series [La1, Chapter7]. Therefore L2

res(Γ\S) is called the residual subspace.Let Ndis

Γ (λ), N cusΓ (λ), and N res

Γ (λ) be the counting function of theeigenvalues with eigenfunctions belonging to the corresponding subspaces.The following general results about the growth of the counting func-tions are known for any lattice Γ in a real semi-simple Lie group. Letn = dim S. Donnelly [Do] has proved the following bound for the cusp-idal spectrum

lim supλ→∞

N cusΓ (λ)λn

≤ vol(Γ\S)(4π)n/2Γ

(n2 + 1

) , (5.2)

where Γ(s) denotes the Gamma function. For the full discrete spectrum,we have at least an upper bound for the growth of the counting function.The main result of [Mu2] states that

NdisΓ (λ) ! (1 + λ4n ). (5.3)

This result implies that invariant integral operators are trace class onthe discrete subspace which is the starting point for the trace formula.The proof of (5.3) relies on the description of the residual subspace interms of iterated residues of Eisenstein series. One actually expects thatthe growth of the residual spectrum is of lower order than the cuspidalspectrum. For SL(n) the residual spectrum has been determined byMoeglin and Waldspurger [MW]. Combined with (5.2) it follows thatfor G = SL(n) we have

N resΓ(N )(λ) ! λd−1 , (5.4)

where d = dim SL(n, R)/SO(n).

158 Werner Müller

In [Sa2] Sarnak conjectured that if rk (G/K) > 1, each irreduciblelattice Γ in G is essentially cuspidal in the sense that Weyl’s law holdsfor N cus

Γ (λ), i.e., equality holds in (5.2). This conjecture has now beenestablished in quite generality. A. Reznikov proved it for congruencegroups in a group G of real rank one, S. Miller [Mi] proved it for G =SL(3) and Γ = SL(3, Z), the author [Mu3] established it for G = SL(n)and a congruence group Γ. The method of [Mu3] is an extension of theheat equation method described in the previous section for the case ofthe upper half-plane. More recently, Lindenstrauss and Venkatesh [LV]proved the following result.

Theorem 5.1 Let G be a split adjoint semi-simple group over Q andlet Γ ⊂ G(Q) be a congruence subgroup. Let n = dim S. Then

N cusΓ (λ) ∼ vol(Γ\S)

(4π)n/2Γ(

n2 + 1

)λn , λ →∞.

The method is based on the construction of convolution operators withpure cuspidal image. It avoids the delicate estimates of the contributionsof the Eisenstein series to the trace formula. This proves existence ofmany cusp forms for these groups.

The next problem is to estimate the remainder term. For G = SL(n),this problem has been studied by E. Lapid and the author in [LM].Actually, we consider not only the cuspidal spectrum of the Laplacian,but the cuspidal spectrum of the whole algebra of invariant differentialoperators.

As D(S) preserves the space of cusp forms, we can proceed as in thecompact case and decompose L2

cus(Γ\S) into joint eigenspaces of D(S).Given λ ∈ a∗C/W , let

Ecus(λ) =ϕ ∈ L2

cus(Γ\S) : Dϕ = χλ(D)ϕ

be the associated eigenspace. Each eigenspace is finite-dimensional. Letm(λ) = Ecus(λ). Define the cuspidal spectrum Λcus(Γ) to be

Λcus(Γ) = λ ∈ a∗C/W : m(λ) > 0.

Then we have an orthogonal direct sum decomposition

L2cus(Γ\S) =

⊕λ∈Λcus(Γ)

Ecus(λ).

Let the notation be as in (2.18) and (2.19). Then in [LM] we established


the following extension of main results of [DKV] to congruence quotientsof S = SL(n, R)/SO(n).

Theorem 5.2 Let d = dimS. Let Ω ⊂ a∗ be a bounded domain withpiecewise smooth boundary. Then for N ≥ 3 we have∑λ∈Λcus(Γ(N ))

λ∈itΩ

m(λ) =vol(Γ(N)\S)

|W |

∫tΩ

β(iλ) dλ+O(td−1(log t)max(n,3)

),

(5.5)as t →∞, and ∑

λ∈Λcus(Γ(N ))λ∈Bt (0)\ia∗

m(λ) = O(td−2) , t→∞. (5.6)

If we apply (5.5) and (5.6) to the unit ball in a∗, we get the followingcorollary.

Corollary 5.3 Let G = SL(n) and let Γ(N) be the principal congruencesubgroup of SL(n, Z) of level N . Let S = SL(n, R)/SO(n) and d =dim S. Then for N ≥ 3 we have

N cusΓ(N )(λ) =

vol(Γ(N)\S)(4π)d/2Γ

(d2 + 1

)λd + O(λd−1(log λ)max(n,3)

), λ →∞.

The condition N ≥ 3 is imposed for technical reasons. It guaranteesthat the principal congruence subgroup Γ(N) is neat in the sense ofBorel, and in particular, has no torsion. This simplifies the analysis byeliminating the contributions of the non-unipotent conjugacy classes inthe trace formula.

Note that Λcus(Γ(N)) ∩ ia∗ is the cuspidal tempered spherical spec-trum. The Ramanujan conjecture [Sa3] for GL(n) at the Archimedeanplace states that

Λcus(Γ(N)) ⊂ ia∗

so that (5.6) is empty, if the Ramanujan conjecture is true. However,the Ramanujan conjecture is far from being proved. Moreover, it isknown to be false for other groups G and (5.6) is what one can expectin general.

The method to prove Theorem 5.2 is an extension of the methodof [DKV]. The Selberg trace formula, which is one of the basic tools in[DKV], needs to be replaced by the Arthur trace formula [A1], [A2]. This

160 Werner Müller

requires to change the framework and to work with the adelic setting.It is also convenient to replace SL(n) by GL(n).

Again, one of the main issues is to estimate the terms in the traceformula which are associated to Eisenstein series. Roughly speaking,these terms are a sum of integrals which generalize the integral∫ ∞

−∞h(r)

ϕ′

ϕ(1/2 + ir) dr

in (4.1). The sum is running over Levi components of parabolic sub-groups and square integrable automorphic forms on a given Levi com-ponent. The functions which generalize ϕ(s) are obtained from the con-stant terms of Eisenstein series. In general, they are difficult to de-scribe. The main ingredients are logarithmic derivatives of automorphicL-functions associated to automorphic forms on the Levi components.As example consider G = SL(3), Γ = SL(3, Z), and a standard maximalparabolic subgroup P which has the form

P =(

m1 X

0 m2

) ∣∣∣ mi ∈ GL(ni, R), det m1 · det m2 = 1

,

with n1+n2 = 3. Thus there are exactly two standard maximal parabolicsubgroups. The standard Levi component of P is

L =(

m1 00 m2

) ∣∣∣ mi ∈ GL(ni, R), detm1 · det m2 = 1

.

So L is isomorphic to GL(2, R). The Eisenstein series are associatedto Maass cusp forms on Γ(1)\H. The constant terms of the Eisensteinseries are described in [Go, Proposition 10.11.2]. Let f be a Maass cuspform for Γ(1) and let Λ(s, f × f) be the completed Rankin-Selberg L-function associated to f (cf. [Bu]). Then the relevant constant term ofthe Eisenstein series associated to f is given by

Λ(s, f × f)Λ(1 + s, f × f)

.

To proceed one needs a bound similar to (4.6). Assume that ∆f =(1/4 + r2)f . Using the analytic properties of Λ(s, f × f) one can showthat for T ≥ 1 ∫ T

−T

Λ′

Λ(1 + it, f × f) dt ! T log(T + |r|). (5.7)

This is the key result that is needed to deal with the contribution of theEisenstein series to the trace formula.


The example demonstrates a general feature of spectral theory onlocally symmetric spaces. Harmonic analysis on higher rank spacesrequires the knowledge of the analytic properties of automorphic L-functions attached to cusp forms on lower rank groups. For GL(n), thecorresponding L-functions are Rankin-Selberg convolutions L(s, ϕ1×ϕ2)of automorphic cusp forms on GL(ni), i = 1, 2, where n1 + n2 = n

(cf. [Bu], [Go] for their definition). The analytic properties of theseL-functions are well understood so that estimates similar to (5.7) canbe established. For other groups G (except for some low dimensionalcases) our current knowledge of the analytic properties of the corre-sponding L-functions is not sufficient to prove estimates like (5.7). Onlypartial results exist [CPS]. This is one of the main obstacles to extendTheorem 5.2 to other groups.

Bibliography[A1] J. Arthur, A trace formula for reductive groups I: terms associated to

classes in G(Q), Duke. Math. J. 45 (1978), 911–952.[A2] J. Arthur, A trace formula for reductive groups II: applications of a trun-

cation operator, Comp. Math. 40 (1980), 87–121.[Av] V. G. Avakumović, Über die Eigenfunktionen auf geschlossenen Rie-

mannschen Mannigfaltigkeiten. Math. Z. 65 (1956), 327–344.[Bo] A. Borel, Linear algebraic groups, Second edition. Graduate Texts in

Mathematics, 126. Springer-Verlag, New York, 1991.[BG] A. Borel, H. Garland, Laplacian and the discrete spectrum of an arith-

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[CV] Y. Colin de Verdiere, Pseudo-Laplacians II, Ann. Inst. Fourier, Grenoble,33 (1983), 87–113.

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[DG] J.J. Duistermaat, V. Guillemin, The spectrum of positive elliptic opera-tors and periodic bicharacteristics. Invent. Math. 29 (1975), no. 1, 39–79.

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Cambridge Studies in Advanced Mathematics, 99. Cambridge UniversityPress, Cambridge, 2006.

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8Weyl’s Lemma, One of Many

Daniel W. Stroock1

M.I.T.2-272, Cambridge, MA 02140, U.S.A.

[email protected]

Abstract This note is a brief, and somewhat biased, account of theevolution of what people working in P.D.E.’s call Weyl’s Lemma aboutthe regularity of solutions to second order elliptic equations. As dis-tinguished from most modern treatments, which are based on pseudod-ifferential operator technology, the approach adopted here is potentialtheoretic, like Weyl’s own.

1 Where It All Started

Given a bounded, connected open region Ω ⊆ RN with smooth boundary∂Ω and a smooth Φ : Ω −→ RN , consider the problem of smoothlydecomposing Φ into a divergence free part Φ0 and an exact part Φ1 .That is, the problem of writing Φ = Φ0 + Φ1 , where Φ0 and Φ1 aresmooth, divΦ0 vanishes, and Φ1 = ∇ϕ for some ϕ which vanishes at ∂Ω.

To solve this problem, one should begin by observing that, if oneignores questions of smoothness, then it is reasonably clear how to pro-ceed. Namely, because Φ0 is divergence free, and Φ1 is exact, divΦ0 = 0and Φ1 = ∇ϕ. Hence, if ϕ vanishes at ∂Ω, the divergence theorem saysthat ∇ϕ is perpendicular to Φ0 in L2(Ω; RN ). With this in mind, letΦ1 denote the orthognal projection in L2(Ω; RN ) of Φ onto the closurein L2(Ω; RN ) of ∇ψ : ψ ∈ C∞

c (Ω; RN ). Next, choose ϕn : n ≥ 0 ⊆C∞

c (Ω; R) so that ∇ϕn −→ Φ1 in L2(Ω; RN ). Because the ϕn ’s vanish at∂Ω, λ0‖ϕn − ϕm‖2

L2 (Ω;R) ≤ ‖∇ϕn − ϕm‖2L2 (Ω;RN ) , where −λ0 < 0 is the

largest Dirichlet eigenvalue of Laplacian ∆ on L2(Ω; R). Hence, there isa ϕ ∈ L2(Ω; R) to which the ϕn ’s converge. Morever, if ψ ∈ C∞

c (Ω; R),

1 The author acknowledges support from NSF grant DMS 0244991.

164

Weyl’s Lemma, one of many 165

then∫Ω

∆ψ(x)ϕ(x) dx = limn→∞

∫Ω

∆ψ(x)ϕn (x) dx

= − limn→∞

∫Ω∇ψ(x) · ∇ϕn (x) dx

= −∫

Ω∇ψ(x) · Φ1(x) dx

= −∫

Ω∇ψ(x) · Φ(x) dx =

∫Ω

ψ(x)divΦ(x) dx

That is, ∆ϕ = divΦ in the sense of (Schwartz) distributions.In view of the preceding, we will be done as soon as we show that ϕ is

smooth. Indeed, if ϕ is smooth, then ∆ϕ = div(Φ) in the classical sense,and so, when Φ1 = ∇ϕ, Φ0 ≡ Φ−∇ϕ satisfies div(Φ0) = div(Φ)−∆ϕ =0.

2 Weyl’s Lemma

As we saw in §1, the problem posed there will be solved as soon as weshow that ϕ is smooth, and it is at this point that Weyl made a crucialcontribution. Namely, he proved (cf. [6]) the following statement.

Lemma 2.1 (Weyl’s Lemma) Let Ω ⊆ RN be open. If u ∈ D′(Ω; R)(the space of Schwartz distributions on Ω) satisfies ∆u = f ∈ C∞(Ω; R)in the sense that

〈∆ψ, u〉 = 〈ψ, f〉, ψ ∈ C∞c (Ω; R),

then u ∈ C∞(Ω; R).

Proof : Set

γt(x) = (4πt)−N2 exp

[−|x|

2

4t

].

Given x0 ∈ Ω, choose r > 0 so that B(x0 , 3r) ⊂⊂ Ω and

η ∈ C∞c(B(x0 , 3r); [0, 1]

)so that η = 1 on B(x0 , 2r).

Set v = ηu and w = ∆v − ηf . Then w is supported in B(x0 , 3r) \B(x0 , 2r). Now take

vt(x) = γt v(x) ≡ 〈γt(· − x), v〉 and wt(x) = γt w ≡ 〈γt(· − x), w〉.

166 Daniel W. Stroock

For each t > 0, vt is smooth. Moreover,

vt(x) = 〈γt(· − x), ηf〉+ 〈γt(· − x), w〉,

and so

vt(x) = v1(x)−∫ 1

t

γτ (ηf)(x) dτ −∫ 1

t

wτ (x) dτ.

The first term on the right causes no problems, since

ηf ∈ C∞c(B(x0 , 3); R

).

Finally,

sup(τ ,x)∈(0,1)×B (x0 ,r)

∣∣∂αwτ (x)∣∣ < ∞

for all α ∈ NN . Hence, we have now shown that every derivative ofvt B(x0 , 1) is uniformly bounded by a bound which is independentof t ∈ (0, 1]. Since, as t 0, vt −→ v on B(x0 , 1) in the sense ofdistributions, this means that v B(x0 , 1) is smooth.

So far as I know, Weyl’s Lemma is the first definitive statement ofwhat are now known as elliptic regularity results. More precisely, it isthe statement that ∆ is hypoelliptic in the sense that the singular supportof a distribution u is contained in that of ∆u.

The spirit of Weyl’s own proof is very much like that of the one justgiven. Namely, it is based on an analysis of the singularity in the Green’sfunction. The only difference is that he dealt with the Green’s functiondirectly, whereas we have used the mollification of the Green’s functionprovided by the heat flow. Most modern proofs of hypoellipticity provea more quantitative statement. Namely, they prove hypoellipticity as aconsequence of a elliptic estimate which says that the s-order Sobolevnorm (I −∆)u can be used to dominate the (s + 2)-order Sobolev normof u.

3 Weyl’s Lemma for Heat Equation

As we will see, there are various directions in which Weyl’s Lemmahas been extended. The following is an important example of such anextension.

Theorem 3.1 (Weyl’s Lemma for the Heat Equation) Let Ω ⊆ R1×RN

be open. If u ∈ D′(Ω; R) satisfies (∂ξ + ∆)u = f ∈ C∞(Ω; R) in the


sense that

〈(−∂ξ + ∆)ψ, u〉 = 〈ψ, f〉, ψ ∈ C∞c (Ω; R),

then u ∈ C∞(Ω; R).

Proof Given (ξ0 , x0) ∈ Ω, set P (r) = (ξ0−r, ξ0+r)×B(x0 , r), and chooser > 0 so that P (3r) ⊂⊂ Ω. Next, choose η ∈ C∞

c(P (3r); [0, 1]

)so that

η = 1 on P (2r), and set v = ηu and w = (∂ξ + ∆)v − ηf . Also, chooseρ ∈ C∞

c((2, 3); [0,∞)

)with total integral 1, and set ρt(x) = t−1ρ(t−1x).

Finally, for t ∈ (0, 1], set

vt(ξ, x) = 〈ρt(· − ξ)γ·−ξ (∗ − x), v〉 and

wt(ξ, x) = 〈ρt(· − ξ)γ·−ξ (∗ − x), w〉 .

Because ddt ρt(ξ) = −ρ′t(ξ), where ρ(ξ) = ξρ(ξ) and ρt(ξ) = t−1 ρ(t−1ξ),

d

dtvt(ξ, x) = −〈ρ′t(· − ξ)γ·−ξ (∗ − x), v〉

= 〈ρt(· − ξ)γ·−ξ (∗ − x), ηf〉+ wt(ξ, x).

The first term causes no problem as t 0 because ηf ∈ C∞c (Ω; R)

and ρ ∈ L1(R). As for the second term, so long as (ξ, x) ∈ P (r), itsderivatives are controlled independent of t ∈ (0, 1] because

(i) supp(w) ⊆ P (3r) \ P (2r).(ii) Derivatives of ρt are bounded by powers of t−1 .

(iii) For (ξ, x) ∈ P (r) and (ξ′, x′) /∈ P (2r) with 0 < ξ′ − ξ < 3t, allderivatives of γξ ′−ξ (x′ − x) are bounded uniformly by any powerof t.

Thus, just as before, we can conclude that v ∈ C∞(P (r); R

).

4 A General Result

If one examines the proofs given in §§ 2 & 3, one sees that they turnon two properties of the classic heat flow. The first of these is that theheat flow does “no damage” to initial data. That is, if one starts withsmooth data, then it evolves smoothly. The second property is thatso long as one stays away from the diagonal, the heat kernel γt(y − x)remains smooth as t 0. When dealing with ∆, these two suffice.When dealing with ∆ + ∂ξ , one needs a more quantitative statementof the latter property. Namely, one needs to know that away from the


diagonal, the heat kernel goes to 0 faster than any power of t. Based onthis discussion, we formulate the following general principle.Suppose that L is a linear partial differential operator from C∞(RN ; R)

to itself, and assume that associated with L there is a kernel

(t, x, y) ∈ (0, 2)× RN × RN −→ q(t, x, y) ∈ R

and the operators t Qt given by

Qtϕ(x) =∫

ϕ(y)q(t, x, y) dy, ϕ ∈ C∞c (RN ; R)

with the properties that

(i) For each x ∈ RN , (t, y) q(t, x, y) satisfies the adjoint equationwith initial value δx . That is, ∂tq(t, x, y) = [L∗q(t, x, ·)](y) andq(t, x, ·) −→ δx as t 0.

(ii) For each n ≥ 0, there exists a Cn <∞ such that

supt∈(0,2]

‖Qtϕ‖C nb∨ ‖Q∗

t ϕ‖C nb≤ Cn‖ϕ‖C n

b.

Using the same ideas on which we based the proofs in §§ 2 & 3, onecan prove the following.

Theorem 4.1 If, in addition to (i) and (ii), for each n ≥ 0 and ε > 0,

max‖α‖+‖β‖≤n

supt∈(0,1]|y−x|≥ε

∣∣∂αx ∂β

y q(t, x, y)∣∣ <∞,

then L is hypoelliptic. If, for each n ≥ 0, ε > 0, and ν > 0,

maxm+‖α‖+‖β‖≤n

supt∈(0,1]|y−x|≥ε

t−ν∣∣∂m

t ∂αx ∂β

y q(t, x, y)∣∣ <∞,

then L + ∂ξ is hypoelliptic.

5 Elliptic Operators

The original generization of Weyl’s Lemma was to replace the Laplaceoperator by a variable coefficient, second order, elliptic partial differen-tial operator. That is, let a : RN −→ RN ⊗ RN , b : RN −→ RN , andc : RN −→ R be smooth, bounded functions with bounded derivativesof all orders, and set

L =N∑

i,j=1

a(x)ij ∂xi∂xj

+N∑

i=1

b(x)i∂xi+ c(x).


Without loss in generality, we assume that a(x)ij = a(x)j i . The operatorL is said to be uniformly elliptic if a(x) ≥ δI for some δ > 0.

Theorem 5.1 If L is uniformly elliptic, then there is a q ∈ C∞((0,∞)×

RN × RN ; R)

such that ∂tq(t, x, y) = [L∗q(t, x, ·)](y), q(t, x, ·) −→ δx ast 0 for each x ∈ RN , and, for each m ≥ 0 and (α, β) ∈ (NN )2 , thereis a Km,α,β <∞ such that∣∣∂m

t ∂αx ∂β

y q(t, x, y)∣∣ ≤ Km,α,β t−

N + 2 m + ‖α ‖+ ‖β ‖2 exp

[− |y − x|2

Km.α,β t

].

In particular, L + ∂ξ is hypoelliptic.

To see how such a result gets applied, consider the following contruc-tion. Take a and b as in the preceding and c ≡ 0. Let Γ(t, x) denote theGaussian probability measure on RN with mean x+tb(x) and covariance2ta(x). That is, Γ(t, x) has density given by[

(4πt)N det(a(x))]− 1

2 exp[− (y − x− tb(x)) · a(x)−1(y − x− tb(x))

4t

].

Thend

dt

∫ϕ(y) Γ(t, x, dy) =

∫Lxϕ(y) Γ(t, x, dy),

where Lx is the constant coefficient operator obtained by freezing thecoefficients of L at x.

For n ≥ 0, define Pn (t, x) = δx and

Pn (t, x) =∫

Γ(t− [t]n , x′)Pn

([t]n , x, dx′),

where [t]n = 2−n [2n t] is the largest dyadic m2−n dominated by t. Then,

〈ϕ,Pn (t, x)〉 − ϕ(x) =∫ t

0

(∫ ⟨Lyϕ,Γ

(τn , y)

⟩Pn

([τn ], x, dy)

)dτ,

where τn = τ − [τ ]n . Using elementary facts about weak convergenceof probability measures, one can show that there is a continuous map(t, x) P (t, x) such that Pn (t, x) −→ P (t, x) uniformly on compacts.Moreover, because, uniformly on compacts,⟨

Lyϕ,Γ(τn , y)

⟩−→ Lϕ(y) as n →∞,

〈ϕ,P (t, x)〉 = ϕ(x) +∫ t

0〈Lϕ,P (τ, x)〉 dτ.


Thus, ∂tP (t, x) = L∗P (t, x) and P (t, x) −→ δx . Finally, given T > 0,define the distribution u on (0, T )× RN by

〈ϕ, u〉 =∫ T

0

(∫ϕ(y)P (T − t, x, dy)

)dt.

Then (L + ∂ξ )u = 0, and so u is smooth on (t, y) : a(y) > 0. That is,P (t, x, dy) = p(t, x, y)dy where (t, y) p(t, x, y) is a smooth functionthere. Similar reasoning shows that (t, x, y) p(t, x, y) is smooth on(t, x, y) : a(x) > 0 & a(y) > 0, and more delicate considerations showthat it is smooth on (t, x, y) : a(x) > 0 or a(y) > 0.

6 Kolmogorov’s Example

Although ellipticity guarantees hypoellipticity, hypoellipticity holds infor many operators which are not elliptic. The following example due toKolmogorov is seminal.

Take N = 2, and consider L = ∂2x1

+ x1∂x2 , which is severely non-elliptic. As Kolmogorov realized, the corresponding diffusion has coor-dinates

X1(t) = x1 +√

2 B(t) & X2(t) = x2 +∫ t

0X1(τ) dτ,

where t B(t) is a standard, 1-dimensional Brownian motion. In par-ticular, this means that the distribution of

(X1(t),X2(t)

)is the Gaussian

measure on R2 with mean

m(t, x) =(

x1

x2 + tx2

)and covariance

C(t) =

(2t t2

t2 2t3

3 .

)Thus, the fundamental solution to the heat equation ∂tu = Lu is

q(t, x, y) =1

2π√

detC(t)×exp

[−(y −m(t, x)

)· C(t)−1

(y −m(t, x)

)2

].

In particular, L + ∂ξ is hypoelliptic. One can use the same reasoning todraw the same conclusion about when

L = ∂2x1

+N∑

i=2

xi−1∂xi.


7 Results of Hörmander Type

Kolmogorov’s example was put into context by a remarkable resultproved by Hörmander. To state his result, let X0 , . . . , Xr be a setof smooth vector fields on RN and set

L = X0 +r∑

i=1

X2i ,

where the Xk ’s are interpreted as directional derivative operators andX2

k is the composition of Xk with itself. Equivalently, if

Xi(x) =N∑

j=1

σ(x)ij ∂xj, 1 ≤ i ≤ r, and X0(x) =

N∑j=1

βj (x)∂xj,

then the matrix a of second order coefficients equals σσ and the vectorb of first order coeficient part equalsβ1(x)

...βN (x)

+r∑

i=1

Xiσi1(x)...

XiσiN (x)

.

Let L and L′ denote the Lie algebras generated by, respectively,

X0 , . . . , Xr

and

[X0 ,X1 ], . . . , [X0 ,Xr ],X1 , . . . , Xr.

Hörmander proved the following result in [1].

Theorem 7.1 (Hörmander’s Theorem) If L(x) has dimension N at eachx ∈ Ω, then L is hypoelliptic on Ω. If L′(x) has dimension N at eachx ∈ Ω, then ∂ξ + L is hypoelliptic in R1 × Ω.

First Oleinik and Radekevich [5] and later (see [2]) Fefferman, Phong,and others extended and sharpened this theorem to cover cases when L

cannot be represented in terms of vector fields. That is, when there isno smooth square root of a. There are situations in which more detailedinformation is available. For instance, suppose that X0 =

∑r1 ckXk for

some smooth c1 , . . . , ck with bounded derivatives of all orders and thatthe vector fields Xk have bounded derivatives of all orders. Further,


assume that there is an n ∈ N and ε > 0 such thatn∑

m=1

∑α∈1,...,rm

(Vα (x), ξ

)2RN ≥ ε|ξ|2 , (x, ξ) ∈ RN × RN ,

where, for α ∈ 1, . . . , rm , Vα (x) = Xαx and, for m ≥ 2,

Xα =[Xαm

,Xα ′]

when α′ = (α1 , . . . , αm−1).

Then Rothschield and Stein showed that the operator L can be inter-preted in terms of a degenerate Riemannian geometry in which the modelspace is a nilpotent Lie group instead of Euclidean space. Variations onand extensions of their ideas can be found in [2] and [4].

8 Concluding Remarks

One can show that if there is smooth differentiable manifold M for whichL(x) is the tangent space at each x ∈ M , then a diffusion process gen-erated by L which starts on at an x ∈ M will stay on M for a positivelength of time. As a consequence, one can see that when such a manifoldexists, L cannot be hypoellictic in a neighborhood of M . Thus, when onecombines (cf. § 7 in part II of [4]) this with Nagumo’s Theorem aboutintegral manifolds for real analytic vector fields, one realizes that thecriterion in Hörmander’s Theorem is necessary and sufficient when theXk ’s are real analytic.

When the Xk ’s are not real analytic, Hörmander’s criterion is nec-essary and sufficient for subellipticiy but not for hypoellipticity, Forexample, take N = 3 and consider

L = ∂2x1

+(α(x1)∂x2

)2 + ∂2x3

,

where α is a smooth function which vanishes only at 0 but vanishes toall orders there. Further, assume that α2 is an even function on R whichis non-decreasing on [0,∞). Then (cf. the last part of § 8 in part II of[4]) L is hypoelliptic in a neighborhood of 0 if and only if

limξ0

ξ log(|α(ξ)|

)= 0.

When dealing with elliptic operators, hypoellipticity extends easily tosystems. However, the validity of a Hörmander type theorem for systemsremains an open question. Indeed, there is considerable doubt aboutwhat criterion replaces Hörmander’s for systems. Recently, J.J. Kohn [3]has made some progress in this direction. Namely, he has found examples


of complex vector fields for which hypoellipticity holds in the absenceof ellipticity. Perhaps the most intriguing aspect of Kohn’s example isthat, from a subelliptic standpoint, his operators “lose derivatives.”

Bibliography[1] Hörmander, L., Hypoelliptic second order differential operators, Acta.

Math., vol. 119 (1967), 147–171[2] Jerison, D. and Sánchez-Calle A., Subelliptic second-order differential op-

erators, Lecture Notes in Math. #1277 (1986), 46–77[3] Kohn, J.J., Hypoellipticity with loss of derivatives, Ann. of Math.vol. 162

# 2 (2005), 943-986[4] Kusuoka, S. and Stroock, D., Applicatons of the Malliavin calculus, II &

III, J. Fac. Sci. of Tokyo Univ., Sec IA vol. 32 & 34 #1 & #2 (1985 &1987), 1–76 & 391–442

[5] Oleinik, O.A. and Radekevich, E.V., Second Order Equations with Non-negative Characteristic Form, Plenum, NY (1973)

[6] Weyl, H., The method of orthogonal projection in potential theory, DukeMath. J., vol. 7 (1940), 414–444

9Analogies between analysis on foliated spaces

and arithmetic geometryChristopher DeningerMathematisches Institut

Einsteinstr. 6248149 Münster, Germany

[email protected]

1 Introduction

For the arithmetic study of varieties over finite fields powerful cohomo-logical methods are available which in particular shed much light on thenature of the corresponding zeta functions. For algebraic schemes overspec Z and in particular for the Riemann zeta function no cohomologytheory has yet been developed that could serve similar purposes. Fora long time it had even been a mystery how the formalism of such atheory could look like. This was clarified in [D1]. However until now theconjectured cohomology has not been constructed.There is a simple class of dynamical systems on foliated manifolds whosereduced leafwise cohomology has several of the expected structural prop-erties of the desired cohomology for algebraic schemes. In this analogy,the case where the foliation has a dense leaf corresponds to the casewhere the algebraic scheme is flat over spec Z e.g. to spec Z itself. Inthis situation the foliation cohomology which in general is infinite di-mensional is not of a topological but instead of a very analytic nature.This can also be seen from its description in terms of global differentialforms which are harmonic along the leaves. An optimistic guess wouldbe that for arithmetic schemes X there exist foliated dynamical systemsX whose reduced leafwise cohomology gives the desired cohomology ofX . If X is an elliptic curve over a finite field this is indeed the case withX a generalized solenoid, not a manifold, [D3].We illustrate this philosophy by comparing the “explicit formulas” inanalytic number theory to a transversal index theorem. We also givea dynamical analogue of a recent conjecture of Lichtenbaum on specialvalues of Hasse–Weil zeta functions by using the Cheeger-Müller the-orem equating analytic and Reidemeister torsion. As a new example

174

Analysis on foliated spaces and arithmetic geometry 175

we point out an analogy between properties of Cramer’s function madefrom zeroes of the Riemann zeta function and properties of the trace ofa wave operator. Incidentally, in all cases analytic number theory sug-gests questions in the theory of partial differential operators on foliatedmanifolds.Since the entire approach is not yet in a definitive state our style is de-liberately a bit brief in places. For more details we refer to the referencesprovided.Hermann Weyl was very interested in number theory and one of the pi-oneers in the spectral theory of differential operators. In particular thetheory of zeta-regularized determinants and their application to analytictorsion evolved from his work. In my opinion the analogies reviewed inthe present note and in particular the analytic nature of foliation co-homology show that a deeper understanding of number theoretical zetafunctions will ultimately require much more analysis of partial differen-tial operators and of Weyl’s work than presently conceived.

I would like to thank the referee very much for thoughtful comments.

2 Gradient flows

We introduce zeta functions and give some arithmetic motivation for theclass of dynamical systems considered in the sequel.

Consider the Riemann zeta function

ζ(s) =∏p

(1− p−s)−1 =∞∑

n=1

n−s for Re s > 1 .

It has a holomorphic continuation to C \ 1 with a simple pole ats = 1. The famous Riemann hypothesis asserts that all zeroes of ζ(s)in the critical strip 0 < Re s < 1 should lie on the line Re s = 1/2. Anatural generalization of ζ(s) to the context of arithmetic geometry isthe Hasse–Weil zeta function ζX (s) of an algebraic scheme X/Z

ζX (s) =∏

x∈|X|(1−Nx−s)−1 for Re s > dimX .

Here |X | is the set of closed points x of X and Nx is the number ofelements in the residue field κ(x) of x. For the one-dimensional schemeX = spec Z we recover ζ(s) and for X = spec oK where oK is the ring ofintegers in a number field we obtain the Dedekind zeta function ζK (s).It is expected that ζX (s) has a meromorphic continuation to all of C.This is known in many interesting cases but by no means in general.

176 Christopher Deninger

If X has characteristic p then the datum of X over Fp is equivalent tothe pair (X ⊗Fp , ϕ) where ϕ = F⊗idFp

is the Fp -linear base extension ofthe absolute Frobenius morphism F of X . For example, the set of closedpoints |X | of X corresponds to the set of finite ϕ-orbits o on the Fp -valuedpoints (X⊗Fp)(Fp) = X (Fp) of X⊗Fp . We have log Nx = |o| log p underthis correspondence. Pairs (X ⊗ Fp , ϕ) are roughly analogous to pairs(M,ϕ) where M is a smooth manifold of dimension 2 dimX and ϕ is adiffeomorphism of M . A better analogy would be with Kähler manifoldsand covering self maps of degree greater than one. See remark 2.1 below.

The Z-action on M via the powers of ϕ can be suspended as followsto an R-action on a new space. Consider the quotient

X = M ×pZ R∗+ (2.1)

where the subgroup pZ of R∗+ acts on M × R∗

+ as follows:

pν · (m,u) = (ϕ−ν (m), pν u) for ν ∈ Z,m ∈ M,u ∈ R∗+ .

The group R acts smoothly on X by the formula

ϕt [m,u] = [m, etu] .

Here ϕt is the diffeomorphism of X corresponding to t ∈ R. Note thatthe closed orbits γ of the R-action on X are in bijection with the finiteϕ-orbits o on M in such a way that the length l(γ) of γ satisfies therelation l(γ) = |o| log p.

Thus in an analogy of X/Fp with X, the closed points x of X cor-respond to the closed orbits of the R-action ϕ on X and Nx = p|o|

corresponds to el(γ ) = p|o|. Moreover if d = dimX then dim X = 2d+1.The system (2.1) has more structure. The fibres of the natural pro-

jection of X to R∗+/pZ form a 1-codimensional foliation F (in fact a

fibration). The leaves (fibres) of F are the images of M under the im-mersions for every u ∈ R∗

+ sending m to [m,u]. In particular, the leavesare transversal to the flow lines of the R-action and ϕt maps leaves toleaves for every t ∈ R.

Now the basic idea is this: If X/Z is flat, there is no Frobenius andhence no analogy with a discrete time dynamical system i.e. an actionof Z. However one obtains a reasonable analogy with a continuous timedynamical system by the following correspondence


Dictionary, part 1X d-dimensional regular algebraicscheme over spec Z

triple (X,F , ϕt ), where X is a2d+1-dimensional smooth manifoldwith a smooth R-action ϕ : R ×X → X and a one-codimensionalfoliation F . The leaves of Fshould be transversal to the R-orbits and every diffeomorphism ϕt

should map leaves to leaves.

closed point x of X closed orbit γ of the R action

Norm Nx of closed point x Nγ = exp l(γ) for closed orbit γ

Hasse–Weil zeta functionζX (s) =

∏x∈|X|(1 − Nx−s )−1

Ruelle zeta function (4.3)ζX (s) =

∏γ (1 − Nγ−s )±1

(if the product makes sense)

X → spec Fp triples (X,F , ϕt ) where F is givenby the fibres of an R-equivariant fi-bration X → R∗

+ /pZ

Remark 2.1 A more accurate analogy can be motivated as follows: If(M,ϕ) is a pair consisting of a manifold with a self covering ϕ of degreedeg ϕ ≥ 2 one can form the generalized solenoid M = lim←−(. . .

ϕ−→ Mϕ−→

M −→ . . .), cf. [CC] 11.3. On M the map ϕ becomes the shift isomor-phism and as before we may consider a suspension X = M×pZ R∗

+ . Thisexample suggests that more precisely schemes X/Z should correspondto triples (X,F , ϕt) where X is a 2d+1 dimensional generalized solenoidwhich is also a foliated space with a foliation F by Kähler manifolds ofcomplex dimension d. It is possible to do analysis on such generalizedspaces cf. [CC]. Analogies with arithmetic are studied in more detail in[D2] and [Le].

Construction 2.2 Let us consider a triple (X,F , ϕt) as in the dictio-nary above, let Yϕ be the vector field generated by the flow ϕt and letTF be the tangent bundle to the foliation. Let ωϕ be the one-form onX defined by

ωϕ |T F = 0 and 〈ωϕ, Yϕ 〉 = 1 .

One checks that dωϕ = 0 and that ωϕ is ϕt-invariant i.e. ϕt∗ωϕ = ωϕ

for all t ∈ R. We may view the cohomology class ψ = [ωϕ ] in H1(X, R)defined by ωϕ as a homomorphism

ψ : πab1 (X) −→ R .

Its image Λ ⊂ R is called the group of periods of (X,F , ϕt).


It is known that F is a fibration if and only if rankΛ = 1. In thiscase there is an R-equivariant fibration X −→ R/Λ whose fibres are theleaves of F . Incidentally, a good reference for the dynamical systems weare considering is [Fa].

Consider another motivation for our foliation setting: For a numberfield K and any f ∈ K∗ we have the product formula where p runs overthe places of K

∏p

‖f‖p = 1 .

Here for the finite places i.e. the prime ideals p we have

‖f‖p = Np−ordpf .

Now look at triples (X,F , ϕt) where the leaves of F are Riemann sur-faces varying smoothly in the transversal direction. In particular wehave dim X = 3 = 2d + 1 where d = 1 = dim(spec oK ). Considersmooth functions f on X which are meromorphic on leaves and havetheir divisors supported on closed orbits of the flow. For compact X itfollows from a formula of Ghys that

∏γ

‖f‖γ = 1 .

In the product γ runs over the closed orbits and

‖f‖γ = (Nγ)−ordγ f

where Nγ = el(γ ) and ordγ f = ordz (f |Fz). Here z is any point on γ

and Fz is the leaf through z.So the foliation setting allows for a product formula where the Nγ arenot all powers of the same number. If one wants an infinitely generatedΛ, one must allow the flow to have fixed points ( = infinite places). Aproduct formula in this more general setting is given in [Ko].

We end this motivational section listing these and more analogies someof which will be explained later.


Dictionary, part 2Number of residue characteristics|char(X )| of X

rank of period group Λ of (X,F , ϕt )

Weil étale cohomology of X Sheaf cohomology of X

Arakelov compactification X = X ∪X∞ where X∞ = X ⊗ R

Triples (X,F , ϕt ) where X is a2d+1-dimensional compactificationof X with a flow ϕ and a 1-codimensional foliation. The flowmaps leaves to leaves, however ϕmay have fixed points on X

X (C)/F∞ , where F∞ denotes com-plex conjugation

set of fixed points of ϕt . Notethat the leaf of F containing a fixedpoint is ϕ-invariant.

spec κ(x) → X for x ∈ |X | Embedded circle i.e. knotR/l(γ)Z → X correspondingto a periodic orbit γ (map t+ l(γ)Zto ϕt (x) for a chosen point x of γ).

Product formula for number fields∏p

‖f‖p = 1Kopei’s product formula [Ko]

Explicit formulas of analytic num-ber theory

transversal index theorem for R-action on X and Laplacian alongthe leaves of F , cf. [D3]

− log |dK /Q| Connes’ Euler characteristicχF (X, µ) for Riemann surface lam-inations with respect to transversalmeasure µ defined by ϕt , cf. [D2]section 4

3 Explicit formulas and transversal index theory

A simple version of the explicit formulas in number theory asserts thefollowing equality of distributions in D′(R>0):

1−∑

ρ

etρ + et =∑

p

log p∑k≥1

δk log p + (1− e−2t)−1 . (3.1)

Here ρ runs over the zeroes of ζ(s) in 0 < Re s < 1, the Dirac distri-bution in a ∈ R is denoted by δa and the functions etρ are viewed asdistributions. Note that in the space of distributions the sum

∑ρ etρ

converges as one sees by partial integration since∑

ρ ρ−2 converges.We want to compare this formula with a transversal index theorem in

geometry.


With our dictionary in mind consider triples (X,F , ϕt) where X is asmooth compact manifold of odd dimension 2d+1, equipped with a one-codimensional foliation F and ϕt is a flow mapping leaves of F to leaves.Moreover we assume that the flow lines meet the leaves transversallyin every point so that ϕ has no fixed points. Consider the (reduced)foliation cohomology:

H•F (X) := Ker dF/Im dF .

Here (A•F (X), dF ) with Ai

F (X) = Γ(X,ΛiT ∗F) is the “de Rham com-plex along the leaves”, (differentials only in the leaf direction). MoreoverIm dF is the closure in the smooth topology of A•

F (X), cf. [AK1].The groups H•

F (X) have a smooth linear R-action ϕ∗ induced by theflow ϕt . The infinitesimal generator θ = limt→0

1t (ϕ

t∗ − id) exists onH•

F (X). It plays a similar role as the Frobenius morphism on étale orcrystalline cohomology.In general, the cohomologies H•

F (X) will be infinite dimensional Fréchetspaces. They are related to harmonic forms. Assume for simplicity thatX and F are oriented in a compatible way and choose a metric gFon TF . This gives a Hodge scalar product on A•

F (X). We define theleafwise Laplace operator by

∆F = dFd∗F + d∗FdF on A•F (X) .

Then we have by a deep theorem by Álvarez López and Kordyukov[AK1]:

H•F (X) = Ker ∆F . (3.2)

Note that ∆F is not elliptic but only elliptic along the leaves of F . Hencethe standard regularity theory of elliptic operators does not suffice for(3.2).

The isomorphism (3.2) is a consequence of the Hodge decompositionproved in [AK1]

A•F (X) = Ker ∆F ⊕ Im dFd∗F ⊕ Im d∗FdF . (3.3)

Consider ∆F as an unbounded operator on the space A•F ,(2)(X) of leaf-

wise forms which are L2 on X. Then ∆F is symmetric and its closure∆F is selfadjoint. The orthogonal projection P of A•

F ,(2)(X) to Ker ∆Frestricts to the projection P of A•

F (X) to Ker ∆F in (3.3). There isalso an L2-version H•

F ,(2)(X) of the reduced leafwise cohomology and itfollows from a result of von Neumann that we have

H•F ,(2)(X) = Ker ∆F .


In order to get a formula analogous to (3.1) we have to take dim X = 3.Then F is a foliation by surfaces so that Hi

F (X) = 0 for i > 2. Thefollowing transversal index theorem is contained in [AK2] and [DS]:

Theorem 3.1 Let (X,F , ϕt) be as above with dim X = 3 such that Fhas a dense leaf. Assume that there is a metric gF on TF such that ϕt

acts with conformal factor eαt . (This condition is very strong and can bereplaced by weaker ones.) Then the spectrum of θ on H1

F ,(2)(X) consistsof eigenvalues. If all periodic orbits of ϕt are non-degenerate we havean equality of distributions in D′(R>0) for suitable (known) signs:

1−∑

ρ

etρ + eαt =∑

γ

l(γ)∑k≥1

±δkl(γ ) . (3.4)

In the sum ρ runs over the spectrum of θ on H1F ,(2)(X) and all ρ satisfy

Re ρ = α2 .

Comparing (3.1) and (3.4) we see again that the primes p and theclosed orbits γ should correspond with log p = l(γ). Since we assumedthat the flow ϕt has no fixed points there is no contribution in (3.4) cor-responding to the term (1 − e−2t)−1 in (3.1). Ongoing work of ÁlvarezLópez and Korduykov leads to optimism that there is an extension ofthe formula in the theorem to the case where ϕt may have fixed pointsand where the right hand side looks similar to (3.1). However reducedleafwise L2-cohomology may have to be replaced by “adiabatic cohomol-ogy”.

The conditions in the theorem actually force α = 0 so that ϕt isisometric. This is related to remark 2. In fact in [D3] using ellipticcurves over finite fields we constructed examples of systems (X,F , ϕt)where X is a solenoidal space and we have α = 1.

For simplicity we have only considered the explicit formula on R>0 .The comparison works also on all of R and among other features onegets a beautiful analogy between Connes’ Euler characteristic χF (X,µ)and − log |dK/Q| cf. [D2] section 4.

4 A conjecture of Lichtenbaum and a dynamical analogue

Consider a regular scheme X which is separated and of finite type overspec Z and assume that ζX (s) has an analytic continuation to s = 0.

Lichtenbaum conjectures the existence of a certain “Weil-étale” co-homology theory with and without compact supports, Hi

c(X , A) and


Hi(X , A) for topological rings A. See [Li1], [Li2]. It should be relatedto the zeta function of X as follows.

Conjecture 4.1 (Lichtenbaum) Let X/Z be as above. Then thegroups Hi

c(X , Z) are finitely generated and vanish for i > 2 dimX + 1.Giving R the usual topology we have

Hic(X , Z)⊗Z R = Hi

c(X , R) .

Moreover, there is a canonical element ψ in H1(X , R) which is functorialin X and such that we have:a The complex

. . .D−→ Hi

c(X , R) D−→ Hi+1c (X , R) −→ . . .

where Dh = ψ ∪ h, is acyclic. Note that D2 = 0 because deg ψ = 1.

b ords=0ζX (s) =∑

i(−1)i i rk Hic(X , Z).

c For the leading coefficient ζ∗X (0) of ζX (s) in the Taylor developmentat s = 0 we have the formula:

ζ∗X (0) = ±∏

i

|Hic(X , Z)tors |(−1)i

/det(H•c (X , R),D, f•) .

Here, fi is a basis of Hic(X , Z)/tors.

Explanation For an acyclic complex of finite dimensional R-vectorspaces

0 −→ V 0 D−→ V 1 D−→ . . .D−→ V r −→ 0 (4.1)

and bases bi of V i a determinant det(V •,D, b•) in R∗+ is defined as

follows: For bases a = (w1 , . . . , wn ) and b = (v1 , . . . , vn ) of a finitedimensional vector space V set [b/a] = detM where vi =

∑j mijwj and

M = (mij ). Thus we have [c/a] = [c/b][b/a].For all i choose bases ci of D(V i−1) in V i and a linearly independent

set ci−1 of vectors in V i−1 with D(ci−1) = ci . Then (ci , ci) is a basis ofV i since (4.1) is acyclic and one defines:

det(V •,D, b•) =∏

i

|[bi/(ci , ci)]|(−1)i

. (4.2)

Note that det(V •,D, b•) is unchanged if we replace the bases bi withbases ai such that |[bi/ai ]| = 1 for all i. Thus the bi could be replaced


by unimodularly or orthogonally equivalent bases. In particular thedeterminant

det(H•c (X , R),D, f•)

does not depend on the choice of bases fi of Hic(X , Z)/tors.

The following formula is obvious:

Proposition 4.2 Let ai and bi be bases of the V i in (4.1). Then wehave:

det(V •,D, b•) = det(V •,D, a•)∏

i

|[bi/ai ]|(−1)i

.

For smooth projective varieties over finite fields using the Weil-étaletopology Lichtenbaum has proved his conjecture in [Li1]. See also [Ge]for generalizations to the singular case. The formalism also works nicelyin the study of ζX (s) at s = 1/2, cf. [Ra]. If X is the spectrum of anumber field, Lichtenbaum gave a definition of “Weil-étale” cohomologygroups using the Weil group of the number field. Using the formula

ζ∗K (0) = −hR

w

he was able to verify his conjecture except for the vanishing of cohomol-ogy in degrees greater than three, [Li2]. In fact, his cohomology doesnot vanish in higher degrees as was recently shown by Flach and Geisserso that some modification will be necessary.

For a dynamical analogue let us look at a triple (X,F , ϕt) as in section2 with X a closed manifold of dimension 2d + 1 and ϕt everywheretransversal to F . We then have a decomposition TX = TF ⊕ T0X

where T0X is the rank one bundle of tangents to the flow lines.In this situation the role of Lichtenbaum’s Weil-étale cohomology is

played by the ordinary singular cohomology with Z or R-coefficients ofX. Note that because X is compact we do not have to worry aboutcompact supports. From the arithmetic point of view we are dealingwith a very simple analogue only!

Lichtenbaum’s complex is replaced by

(H•(X, R),D) where Dh = ψ ∪ h and ψ = [ωϕ ] .

Here ωϕ is the 1-form introduced in construction 3 above.Now assume that the closed orbits of the flow are non-degenerate.


Then at least formally we have:

ζR (s) :=∏γ

(1− e−sl(γ ))−εγ

=∏

i

det∞(s · id− θ | HiF (X))(−1)i + 1

.(4.3)

Here γ runs over the closed orbits, εγ = sgn det(1−Txϕl(γ ) |TxF) for anyx ∈ γ, the Euler product converges in some right half plane and det∞ isthe zeta-regularized determinant. The functions det∞(s · id−θ | Hi

F (X))should be entire.

If the closed orbits of ϕ are degenerate one can define a Ruelle zetafunction via Fuller indices [Fu] and relation (4.3) should still hold. Inthe present note we assume for simplicity that the action of the flowϕ on TF is isometric with respect to gF and we do not insist on thecondition that the closed orbits should be non-degenerate. Then θ haspure eigenvalue spectrum with finite multiplicities on H•

F (X) = Ker ∆•F

by [DS], Theorem 2.6. We define ζR (s) by the formula

ζR (s) =∏

i

det∞(s · id− θ | HiF (X))(−1)i + 1

,

if the individual regularized determinants exist and define entire func-tions. The following result is proved in [D4].

Theorem 4.3 Consider a triple (X,F , ϕt) as above with dim X = 2d+1and compatible orientations of X and F such that the flow acts isomet-rically with respect to a metric gF on TF . For b and c assume that theabove zeta-regularized determinants exist. Then the following assertionshold:a The complex

. . .D−→ Hi(X, R) D−→ Hi+1(X, R) −→ . . .

where Dh = ψ ∪ h with ψ = [ωϕ ] is acyclic.

b ords=0ζR (s) =∑

i(−1)i i rk Hi(X, Z).

c For the leading coefficient in the Taylor development at s = 0 we havethe formula:

ζ∗R (0) =∏

i

|Hi(X, Z)tors |(−1)i

/det(H•(X, R),D, f•) .

Here, fi is a basis of Hi(X, Z)/tors.


Idea of a proof of c We define a metric g on X by g = gF + g0

(orthogonal sum) on TX = TF⊕T0X with g0 defined by ‖Yϕ,x‖ = 1 forall x ∈ X. Using techniques from the heat equation proof of the indextheorem and assuming the existence of zeta-regularized determinantsone can prove the following identity:

ζ∗R (0) = T (X, g)−1 . (4.4)

Here T (X, g) is the analytic torsion introduced by Ray and Singerusing the spectral zeta functions of the Laplace operators ∆i on i-formson X

T (X, g) = exp∑

i

(−1)i i

2ζ ′∆ i (0) . (4.5)

Equation (4.4) is the special case of Fried’s conjecture in [Fr3], see also[Fr1], [Fr2] where the flow has an integrable complementary distribution.Next, we use the famous Cheeger–Müller theorem:

T (X, g) = τ(X, g) cf. [Ch], [M] (4.6)

where τ(X, g) = τ(X, h•) is the Reidemeister torsion with respect to thevolume forms on homology given by the following bases hi of Hi(X, R).Choose orthonormal bases hi of Ker ∆i with respect to the Hodge scalarproduct and view them as bases of Hi(X, R) via Ker ∆i ∼−→ Hi(X, R).Let hi be the dual base to hi .

If we are given two bases a and b of a real vector space we have theformula

τ(X, b•) = τ(X, a•)∏

i

|[bi/ai ]|(−1)i

for any choices of bases ai , bi of Hi(X, R). Let f•, f• be dual bases of

H•(X, Z)/tors resp. H•(X, Z)/tors. Then we have in particular:

τ(X, h•) = τ(X, f•)∏

i

|[hi/fi ]|(−1)i

. (4.7)

Now it follows from the definition of Reidemeister torsion that we have:

τ(X, f•) =∏

i


.

The cap-isomorphism:

_ ∩ [X] : Hj (X, Z) ∼−→ H2d+1−j (X, Z)


now implies that

τ(X, f•) =∏

i

|Hi(X, Z)tors |(−1)i + 1. (4.8)

Next let us look at the acyclic complex

(H•(X, R),D = ψ ∪ _) ∼= (Ker ∆•, ωϕ ∧ _) .

Using the canonical decomposition into bidegrees

Ker ∆n = ωϕ ∧ (Ker ∆n−1F )θ=0 ⊕ (Ker ∆n

F )θ=0

we see that it is isometrically isomorphic to

(M•−1 ⊕M•,D) (4.9)

where Mi = (Ker ∆iF )θ=0 and D(m′,m) = (m, 0). We may choose the

orthonormal basis hi above to be of the form hi = (ωϕ ∧ hi−1 , hi) wherehi is an orthonormal basis of (Ker ∆i

F )θ=0 . For this basis, i.e. (hi−1 , hi)in the version (4.9) it is trivial from the definition (4.2) that

|det(H•(X, R),D, h•)| = 1 .

Using proposition 4.2 for b• = h

• and a• = f

• we find

1 = |det(H•(X, R),D, f•)|∏

i

|[hi/fi ]|(−1)i

. (4.10)

Hence we get∏i


/det(H•(X, R),D, f•)

(4.10)=

∏i

|Hi(X, Z)tors |(−1)i ∏i

|[hi/fi |](−1)i

(4.8)= τ(X, f•)−1

∏i

|[hi/fi |](−1)i + 1 (4.7)= τ(X, h•)−1

(4.6)= T (X, g)−1 (4.4)

= ζ∗R (0) .

The simplest example For q > 1 let X be the circle R/(log q)Zfoliated by points and with R acting by translation. The Ruelle zetafunction is given by

ζR (s) = (1− e−s log q )−1 = (1− q−s)−1 .


The operator θ = d/dx on H0F (X) = C∞(X) defines an unbounded op-

erator on H0F ,(2)(X) = L2(X) with pure eigenvalue spectrum 2πiν/ log q

for ν ∈ Z. It follows from a formula of Lerch e.g. [D1] § 2 that we have

det∞(s · id− θ | H0F (X)) = 1− q−s

and hence formula (4.3) holds in this case.Next, we have ωϕ = dt and hence ψ = [dt] is a generator of H1(X, R).

The complex

. . .D−→ Hi(X, R) D−→ Hi+1(X, R) → . . .

is therefore acyclic. Moreover we have

ords=0ζR (s) = −1 =∑

i

(−1)i i rk Hi(X, Z) .

The leading coefficient ζ∗R (0) is given by ζ∗R (0) = (log q)−1 . We haveHi(X, Z)tors = 0 for all i. As ψ/ log q is a basis of H1(X, Z) the formulaD(1) = ψ = (log q)(ψ/ log q) for 1 ∈ H0(X, R) shows that we have

det(H•(X, R),D, f•) = log q .

This illustrates part c of Theorem 4.3.

5 Cramérs function and the transversal wave equation

In this section we observe a new analogy and mention some furtherdirections of research in analysis suggested by this analogy. Let us writethe non-trivial zeroes of ζ(s) as ρ = 1

2 + γ. In [C] Cramér studied thefunction W (z) which is defined for Im z > 0 by the absolutely and locallyuniformly convergent series

W (z) =∑

Im γ>0

eγz .

According to Cramér the function W has a meromorphic continuation toC− = C \ xi |x ≤ 0 with poles only for z = m log p with m ∈ Z,m = 0and p a prime number. The poles are of first order. If we view thelocally integrable function eγ t of t as a distribution on R the series

Wdis =∑

Im γ>0

eγ t

converges in D′(R). Using the mentioned results of Cramér on W onesees that the singular support of Wdis consists of t = 0 and the numberst = m log p for p a prime and m ∈ Z,m = 0, cf. [DSch] section 1. One


my ask if there is an analytic counterpart to this result in the spirit ofthe dictionary in section 2.

Presently this is not the case although the work [K] is related to theproblem. According to our dictionary we consider a system (X,F , ϕt)with a 3-manifold X. Let us first assume that X is compact and that ϕt

is isometric and has no fixed points. Then we have −θ2 = ∆0 on A•F (X)

where θ = limt→01t (ϕ

t∗−id) is the infinitesimal generator of the inducedgroup of operators ϕt∗ on the Fréchet space A•

F (X). Moreover ∆0 is theLaplacian along the flow lines with coefficients in Λ•T ∗F . Note that it istransversally elliptic with respect to F . Since ∆0 |Ker ∆F = ∆ |Ker ∆F byisometry, it follows from the corresponding result for the spectrum of theLaplacian that the spectrum γ of θ on H1

F ,(2)(X) = Ker ∆1F consists

of eigenvalues and that all γ are purely imaginary. (This proves a partof theorem 5.) For ϕ in D(R) the operator

∫R

ϕ(t)eit|θ | dt on Ker ∆1F

is trace class and the map ϕ → tr∫

Rϕ(t)eit|θ | defines a distribution

denoted by tr(eit|θ | |Ker ∆1F ). Then we have:

2∑

Im γ>0

etγ = tr(eit|θ | |Ker ∆1F )

= tr(eit√

∆ |Ker ∆1F ) in D′(R) .

Here we have assumed that zero is not in the spectrum of θ on Ker ∆1F ,

corresponding to the fact that ζ(1/2) = 0. But this is not important.On functions, i.e. on C∞(X) = A0

F (X) instead of Ker ∆1F the dis-

tributional trace tr(eit√

∆) of the operator eit√

∆ has been extensivelystudied. By a basic result of Chazarain [Cha] the singular support oftr(eit

√∆) consists of t = 0 and the numbers t = ml(γ) for m ∈ Z,m = 0

and l(γ) the lengths of the closed orbits of the geodesic flow. On theother hand by the analogy with Cramérs theorem we expect that the sin-gularities of tr(eit

√∆ |Ker ∆

1F ) are contained in the analogous set where

now γ runs over the closed orbits of the flow ϕt .Based on the work of Hörmander the “big” singularity of tr(eit

√∆) at

t = 0 was analyzed in [DG] proposition 2.1 by an asymptotic expansionof the Fourier transform of tr(eit

√∆). An analogue of this expansion for

the Fourier transform of tr(eit√

∆ |Ker ∆1F ) would correspond well with

asymptotics that can be obtained with some effort from Cramér’s the-ory, except that in Cramér’s theory there also appear logarithmic termscoming from the infinite place. These should also appear in the analysisof tr(eit

√∆ |Ker ∆

1F ) if one allows the flow to have fixed points. Finally

one should drop the condition that the flow acts isometrically. Then


one has to work with tr(eit|θ | | H1F ,(2)(X)) instead of tr(eit

√∆ |Ker ∆

1F ).

Also one can try to prove a Duistermaat–Guillemin trace formula fortr(eit|θ | | H1

F ,(2)(X)) and even if the flow has fixed points.

Bibliography[AK1] J. Álvarez López, Y.A. Kordyukov, Long time behaviour of leafwise

heat flow for Riemannian foliations. Compos. Math. 125 (2001), 129–153[AK2] J. Álvarez López, Y.A. Kordyukov, Distributional Betti numbers of

transitive foliations of codimension one. In: Proceedings on Foliations:Geometry and Dynamics, ed. P. Walczak et al. World Scientific, Singa-pore, 2002, pp. 159–183

[CC] A. Candel, L. Conlon, Foliations I. AMS Graduate studies in Mathe-matics 23, 2000.

[Cha] J. Chazarain, Formules de Poisson pour les variétés riemanniennes. In-vent. Math. 24 (1974), 65–82

[Ch] J. Cheeger, Analytic torsion and the heat equation. Ann. Math. 109(1979), 259–322

[C] H. Cramér, Studien über die Nullstellen der Riemannschen Zetafunktion.Math. Z. 4 (1919), 104–130

[DG] J.J. Duistermaat, V.W. Guillemin, The spectrum of positive ellipticoperators and periodic bicharacteristics. Invent. Math. 29 (1975), 39–79

[D1] C. Deninger, Motivic L-functions and regularized determinants, (1992),in: Jannsen, Kleimann, Serre (eds.): Seattle conference on motives 1991.Proc. Symp. Pure Math. AMS 55 (1994), Part 1, 707–743

[D2] C. Deninger, Number theory and dynamical systems on foliated spaces.Jber. d. Dt. Math.-Verein. 103 (2001), 79–100

[D3] C. Deninger, On the nature of the “explicit formulas” in analytic num-ber theory – a simple example. In: S. Kanemitsu, C. Jia (eds.), NumberTheoretic Methods – Future Trends. In: DEVM “Developments of Math-ematics”, Kluwer Academic Publ.

[D4] C. Deninger, A dynamical systems analogue of Lichtenbaum’s conjec-tures on special values of Hasse–Weil zeta functions. Preprint arXiVmath.NT/0605724

[DS] C. Deninger, W. Singhof, A note on dynamical trace formulas. In: M.L.Lapidus, M. van Frankenhuysen (eds.), Dynamical Spectral and Arith-metic Zeta-Functions. In: AMS Contemp. Math. 290 (2001), 41–55

[DSch] C. Deninger, M. Schröter, A distribution theoretic interpretation ofGuinand’s functional equation for Cramér’s V -function and generaliza-tions. J. London Math. Soc. (2) 52 (1995), 48–60

[Fa] M. Farber, Topology of closed geodesics on hyperbolic manifolds. Math-ematical Surveys and Monographs, Vol. 108, AMS 2004

[Fr1] D. Fried, Homological identities for closed orbits. Invent. Math. 71(1983), 419–442

[Fr2] D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds.Invent. Math. 84 (1986), 523–540

[Fr3] D. Fried, Counting circles. In: Dynamical Systems LNM 1342 (1988),196–215

[Fu] Fuller, An index of fixed point type for periodic orbits. Amer. J. Math.89 (1967), 133–148


[Ge] T. Geisser, Arithmetic cohomology over finite fields and special values ofζ-functions. Duke Math. J. 133 (2006), 27–57

[Ko] F. Kopei, A remark on a relation between foliations and number theory.ArXiv math.NT/0605184

[K] Y. Kordyukov, The trace formula for transversally elliptic operators onRiemannian foliations. St. Petersburg Math. J. 12 (2001), 407–422

[Le] E. Leichtnam, Scaling group flow and Lefschetz trace formula for lami-nated spaces with p-adic transversal. To appear in: Bulletin des SciencesMathématiques

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[RS] D. Ray, I. Singer, Analytic torsion. Proc. Symp. Pure Math. XXIII, 167–181

10Reciprocity Algebras and Branching for

Classical Symmetric PairsRoger E. Howe

Department of MathematicsYale University

Eng-Chye TanDepartment of Mathematics

National University of Singapore

Jeb F. WillenbringDepartment of Mathematical SciencesUniversity of Wisconsin at Milwaukee

Abstract We study branching laws for a classical group G and asymmetric subgroup H. Our approach is by introducing the branchingalgebra, the algebra of covariants for H in the regular functions on thenatural torus bundle over the flag manifold for G. We give concretedescriptions of certain subalgebras of the branching algebra using clas-sical invariant theory. In this context, it turns out that the ten classesof classical symmetric pairs (G,H) are associated in pairs, (G,H) and(H ′, G′).

Our results may be regarded as a further development of classicalinvariant theory as described by Weyl [64], and extended previouslyin [14]. They show that the framework of classical invariant theory isflexible enough to encompass a wide variety of calculations that havebeen carried out by other methods over a period of several decades.This framework is capable of further development, and in some wayscan provide a more precise picture than has been developed in previouswork.

1 Introduction

1.1 The Classical Groups.

Hermann Weyl’s book, The Classical Groups [64], has influenced manyresearchers in invariant theory and related fields in the decades since itwas written. Written as an updating of “classical" invariant theory, it

191

192 R.E. Howe, E.-C. Tan and J.F. Willenbring

has itself acquired the patina of a classic. The books [11] and [55] andthe references in them give an idea of the extent of the influence. Thecurrent authors freely confess to being among those on whom Weyl hashad major impact.

The Classical Groups has two main themes: the invariant theory ofthe classical groups – the general linear groups, the orthogonal groupsand the symplectic groups – acting on sums of copies of their standardrepresentations (and, in the case of the general linear groups, copies ofthe dual representation also); and the description of the irreducible rep-resentations of these groups. In invariant theory, the primary results of[64] are what Weyl called the First and Second Fundamental Theoremsof invariant theory. The First Fundamental Theorem (FFT) describesa set of “typical basic generators" for the invariants of the selected ac-tions, and the Second Fundamental Theorem (SFT) describes the rela-tions between these generators. The description of the representationsculminates in the Weyl Character Formula.

The two themes are not completely integrated. For the first one, Weyluses the apparatus of classical invariant theory, including the Aronholdpolarization operators and the Capelli identity, together with geometri-cal considerations about orbits, etc. For the second, he abandons polyno-mial rings and relies primarily on the Schur-Weyl duality, the remarkableconnection discovered by I. Schur between representations of the generallinear groups and the symmetric groups. This duality takes place on ten-sor powers of the fundamental representation of GLn , which of courseare finite dimensional. This gives Weyl’s description of the irreduciblerepresentations more of a combinatorial cast. This combinatorial view-point, based around Ferrers-Young diagrams and Young tableaux, hasbeen very heavily developed in the latter half of the twentieth century(see [48], [59], [50], [9], [55] and the references below).

1.2 From Invariants to Covariants.

In [14], it was observed that by combining the results in [64] with an-other construction of Weyl, namely the Weyl algebra, aka the algebraof polynomial coefficient differential operators, it is possible to give aunified treatment of the invariants and the irreducible representations.

Introduction of the Weyl algebra brings several valuable pieces ofstructure into the picture. A key feature of the Weyl algebra W(V )associated to a vector space V is that it has a filtration such that theassociated graded algebra is commutative, and is canonically isomorphic

Reciprocity algebras and branching 193

to the algebra P(V ⊕ V ∗) of polynomials on the sum of V with its dualV ∗. Moreover, if one extends the natural bilinear pairing between V

and V ∗ to a skew-symmetric bilinear (or symplectic) form on V ⊕ V ∗,then the symplectic group Sp(V ⊕ V ∗) of isometries of this form actsnaturally as automorphisms of W(V ), and this action gets carried overto the natural action of Sp(V ⊕ V ∗) on P(V ⊕ V ∗). Also, the naturalaction of GL(V ), the general linear group of V , on P(V ⊕ V ∗) embedsGL(V ) in Sp(V ⊕ V ∗). The corresponding action of GL(V ) on W(V )is just the action by conjugation when both GL(V ) and W(V ) are re-garded as being operators on P(V ). Finally, the Lie algebra sp(V ⊕V ∗)of Sp(V ⊕V ∗) is naturally embedded as a Lie subalgebra of W(V ). Theimage of sp(V ⊕V ∗) in P(V ⊕V ∗) consists of the homogeneous polyno-mials of degree two. The Lie bracket is then given by Poisson bracketwith respect to the symplectic form [6].

Given a group G ⊆ GL(V ), it is natural in this context to look atW(V )G , the algebra of polynomial coefficient differential operators in-variant under the action of G, or equivalently, of operators that commutewith G. One can show in a fairly general context that W(V )G providesstrong information about the decomposition of P(V ) into irreduciblerepresentations for G [10].

In the case of the classical actions considered by Weyl, it turns outthat W(V )G has an elegant structure. This structure is revealed byconsidering the centralizer of G inside Sp(V ⊕V ∗). Since G ⊆ GL(V ) ⊂Sp(V ⊕V ∗), we can consider G′ = Sp(V ⊕V ∗)G , the centralizer of G inSp(V ⊕ V ∗). The Lie algebra of G′ will be g′ = sp(V ⊕ V ∗)G , the Liesubalgebra of sp(V ⊕ V ∗) consisting of elements that commute with G.

Given a group S and subgroup H, looking at H ′, the centralizer ofH in S, is a construction that has the formal properties of a dualityoperation, analogous to considering the commutant of a subalgebra inan algebra. It is easy to check that H ′′ = (H ′)′ contains H, and thatH ′′′ = (H ′)′′ is again equal to H ′. Hence also H ′′′′ = H ′′. Thus H ′′

constitutes a sort of “closure" of H with respect to the issue of com-muting inside S, and the pair (H ′′,H ′) constitute a pair of mutuallycentralizing subgroups of S. In [14] such a pair was termed a dual pairof subgroups of S. Dual pairs of subgroups arise naturally in studyingthe structure of groups. For example, in reductive algebraic groups, theLevi component of a parabolic subgroup and its central torus constitutea dual pair. A subgroup H ⊆ G belongs to a dual pair in G if and onlyif it is its own double centralizer.

It turns out that, if G is one of the classical groups acting on V


by one of the actions specified by Weyl, and G′ is its centralizer inSp(V ⊕ V ∗), then (G,G′) constitute a dual pair in Sp(V ⊕ V ∗). Thatis, G = G′′. Moreover, by translating Weyl’s FFT to the context ofthe Weyl algebra, one sees that g′ = sp(V ⊕ V ∗)G generates W(V )G

as an associative algebra. Furthermore, the condition that G = G′′

essentially characterizes the actions considered by Weyl, providing someinsight into why the FFT has such a clean statement for these actions,but is known for hardly any other examples. The fact that g′ generatesW(V )G , together with some simple structural facts about g′, allows adetailed description of the action of G on P(V ). As part of this picture,one obtains a natural bijection between the irreducible representationsof G appearing in P(V ), and certain irreducible representations of g′.

In the case of when G = GLn and V Cn ⊗ Cm , then G′ = GLm

acting on the factor Cm of V . The Lie algebra g′ glm is exactlythe span of the Aronhold polarization operators. In this case, the poly-nomials on V decompose into jointly irreducible representations ρ ⊗ ρ′

for GLn × GLm , and the correspondence ρ ↔ ρ′ is bijective. We referto this correspondence as (GLn,GLm ) duality. The paper [16] further

studied this situation and the foundations of Weyl’s Fundamental The-orems, and pointed out that these results could be understood from thepoint of view of multiplicity free actions. In this development, the FFTfor GLn , Schur-Weyl duality and (GLn,GLm ) duality appear as threeaspects of the same phenomenon. Each can be deduced from either ofthe others. In particular, the polynomial version of the theory and thecombinatorial version are seen as two windows on the same landscape.

1.3 Branching Rules via Invariant Theory.

Already when [64] was being written, work was underway to extend rep-resentation theory beyond a basic enumeration of the irreducible repre-sentations to describe some aspects of their structure. Part of the mo-tivation for doing this came from quantum mechanics, which also wasthe inspiration for the Weyl algebra. In [46], Littlewood and Richardsonproposed a combinatorial description of the multiplicities of irreduciblerepresentations in the tensor product of two irreducible representationsof GLn . These multiplicities are now known as Littlewood-Richardson(LR) coefficients.

The decomposition of tensor products of representations of a group G

can be regarded as a branching problem – the problem of decomposing


the restriction of a representation of some group to a subgroup. Pre-cisely, if ρ1 and ρ2 are irreducible representations of G, then the tensorproduct ρ1⊗ρ2 can be regarded as an irreducible representation of G×G,and all irreducible representations of G×G are of this form [11, 16].

If we restrict this representation to the diagonal subgroup ∆G =(g, g) : g ∈ G ⊂ G × G, then we obtain the usual notion of ρ1 ⊗ ρ2

as a representation of G. Note that ∆G is isomorphic to G. Also, theinvolution (g1 , g2) ↔ (g2 , g1) of G×G has ∆G as the subgroup of fixedpoints. Thus, ∆G is a symmetric subgroup of G × G – the fixed pointset of an involution (order two automorphism) of G × G. We also callthe pair (G×G,∆(G)) a symmetric pair. In summary, computing ten-sor products for representations of GLn can be viewed as studying thebranching problem for the symmetric pair (GLn ×GLn,∆(GLn )).

In the 1940s, Littlewood considered the branching problem from GLn

to the orthogonal group On . Note that On is the set of fixed pointsof the involution g → (g−1)t , where At denotes the transpose of then × n matrix A. Thus, (GLn,On ) is a symmetric pair. Since all therepresentations involved are semisimple (thanks, e.g. to Weyl’s unitar-ian trick [64]), to determine them up to isomorphism, it is enough toknow the branching multiplicities – the multiplicity with which each irre-ducible representation of On occurs in a given irreducible representationof GLn . Under some restrictions on the representations involved, Lit-tlewood [48, 47] showed how to express the branching multiplicities forthe pair (GLn,On ), in terms of LR coefficients. Over the decades since[64], these results have been extended in stages, so that one now hasa description of the branching multiplicities for any classical symmetricpair – a symmetric pair (G,K), where G is a classical group – in termsof LR coefficients [46], [48], [47], [52], [28], [29], [30], [31], [4], [36], [37],[38], [39], [32], [59], [34], [35], [33]. See also [9] and [21].

The goal of this paper is to show how the invariant theory approachcan be further developed to encompass much of the work on branchingmultiplicities cited above. The main ingredient needed for doing this isthe notion of branching algebra, an idea used by Zhelobenko [65], butrelatively little exploited since. The general idea of branching algebra isdescribed in §2.

The concrete description of branching algebras for the classical sym-metric pairs in terms of polynomial rings is carried out in §4. More pre-cisely, in §4, certain families of well behaved subalgebras of branching


algebras are realized via polynomial rings. We refer to these subalgebrasas partial branching algebras.

When one realizes partial branching algebras in the context of dualpairs, a lovely reciprocity phenomenon reveals itself. Suppose that areductive group G ⊂ GL(V ) ⊂ Sp(V ⊕ V ∗) belongs to a dual pair(G,G′). Let K ⊂ G be a symmetric subgroup. It turns out that, ifK ′ ⊂ Sp(V ⊕ V ∗) is the centralizer of K in Sp(V ⊕ V ∗), then (K,K ′)also form a dual pair in Sp(V ⊕ V ∗); and furthermore (K ′, G′) is also aclassical symmetric pair! (Note that, since K ⊂ G, we clearly will haveK ′ ⊃ G′.) Thus, the dual pairs (G,G′) and (K,K ′) form a seesaw pairin the sense of Kudla [41].

Moreover, the partial branching algebra constructed for (G,K) turnsout also to be a partial branching algebra for (K ′, G′)! The coincidenceimplies a reciprocity phenomenon: branching multiplicities for (G,K)also describe branching multiplicities for (K ′, G′). For this reason, wecall these partial branching algebras reciprocity algebras. (It should benoted that often some special infinite dimensional representations of K ′

may be involved in these reciprocity relationships, and sometimes infinitedimensional representations of G′ are also involved.) We note that partsof this picture have appeared before. The fact that (K,K ′) is a dual pair,and that (K ′, G′) is a symmetric pair is implicit in [15]. A numericalversion of the reciprocity laws implied by reciprocity algebras was givenin [13], and the reciprocity phenomenon for the GLn tensor productalgebra was noted in [16].

The classical symmetric pairs may be sorted into ten infinite families(see Table I in §4). It turns out that if the pair (G,K) is taken fromone family, the pair (K ′, G′) is always taken from another family thatis determined by the family of (G,K). That is, the seesaw construc-tion applied to classical symmetric pairs, pairs up the ten families intofive reciprocal pairs of families. The two families in a pair have manyreciprocity laws relating multiplicities of their representations.

Thus, the branching algebra approach to branching rules, when madeconcrete via classical invariant theory, has some highly attractive formalfeatures. We should keep in mind that they are formal, in the sense thatthey don’t say anything explicit about what certain branching multiplic-ities are. They just say that branching multiplicities for one pair (G,K)can be expressed in terms of branching multiplicities for another pair(K ′, G′). To get more specific information, one needs some reasonablyexplicit description of the partial branching algebras. In this paper wegive a relative description, which shows that every reciprocity algebra


is related to the GLn tensor product algebra. This is carried out in §8for one of the symmetric pairs. The theory works best under some tech-nical restrictions, referred to as the stable range. As explained in [21],these relations imply the many formulas in the literature that describebranching multiplicities for symmetric pairs in terms of LR coefficients.

To have a complete theory, one should also give concrete and explicitdescriptions of the reciprocity algebras. This paper does not deal withthis issue. However, it has been carried out, by the authors and others,in several papers. Using ideas of Grobner/SAGBI theory [56], [51], thepaper [22] describes an explicit basis for the basic case, the GLn ten-sor product algebra. Analogous bases for most of the other reciprocityalgebras are given in [17, 18, 19]. Furthermore, the paper [20] showshow to deduce the Littlewood-Richardson Rule from the results of [22]and representation-theoretic considerations. Also, the paper [12] showsthat these branching algebras have toric deformations. More precisely,it shows that they have flat deformations that are semigroup rings oflattice cones, which are explicitly described.

Thus, this paper supplemented by the work just cited shows that theinvariant-theory approach using branching algebras provides a uniformframework to deal with a wide variety of issues in representation theory.These computations have been handled in the literature by largely meansof combinatorics, as cited above, and more recently by quantum groupsand related methods, including the path methods of Littelmann [7], [49],[27], [25], [26], [24], [42], [43], [44], [45]. The branching algebra approachcan provide another window on these phenomena.

The branching algebra approach comes with some extra structure at-tached. The multiplicities are seen not simply as numbers, but intrinsi-cally as cardinalities of integral points in convex sets. (Indeed, one cansee this implicitly in the Littlewood-Richardson Rule, and it was mademore explicit in [2], [3] and [53]).

Moreover, these points do not simply give the correct count: individualpoints correspond to specific highest weight vectors in representations.Finally, the fact that all representations are bundled together inside onealgebra structure implies relations between the highest weight vectorsand the multiplicities of different representations. The authors suspectthat this extra structure can be useful in studying certain problems, suchas the actions analyzed by Kostant and Rallis [40], and perhaps in under-standing the structure of principal series representations of semisimplegroups. We hope to return to these themes in future papers.


Acknowledgements: We thank Kenji Ueno and Chen-Bo Zhu for dis-cussions. The second named author also acknowledges the support ofNUS grant R-146-000-050-112. The third named author was supportedby NSA Grant # H98230-05-1-0078.

2 Branching Algebras

For a reductive complex linear algebraic G, let UG be a maximal unipo-tent subgroup of G. The group UG is determined up to conjugacy inG [5]. Let AG denote a maximal torus which normalizes UG , so thatBG = AG · UG is a Borel subgroup of G. Also let A+

G be the set ofdominant characters of AG – the semigroup of highest weights of rep-resentations of G. It is well-known [5] [16] and may be thought of asa geometric version of the theory of the highest weight, that the spaceof regular functions on the coset space G/UG , denoted by R(G/UG ),decomposes (under the action of G by left translations) as a direct sumof one copy of each irreducible representation Vψ (with highest weightψ) of G (see [60]):

R(G/UG ) ⊕

ψ∈A+G

Vψ . (2.1)

We note that R(G/UG ) has the structure of an A+G -graded algebra,

for which the Vψ are the graded components. To be specific, we notethat since AG normalizes UG , it acts on G/UG by right translations, andthis action commutes with the action of G by left translations.

Proposition 2.1 (see [63]) The algebra of regular functions R(G/UG )is an A+

G -graded algebra, under the right action of AG . More pre-cisely, the decomposition (2.1) is the graded algebra decomposition un-der AG , where Vψ is the AG -eigenspace corresponding to ϕ ∈ A+

G withϕ = w∗(ψ−1). Here w is the longest element of the Weyl group withrespect to the root system determined by the Borel subgroup BG .

Now let H ⊂ G be a reductive subgroup, and let UH be a maximalunipotent subgroup of H. We consider the algebraR(G/UG )UH , of func-tions on G/UG which are invariant under left translations by UH . LetAH be a maximal torus of H normalizing UH , so that BH := AH ·UH isa Borel subgroup of H. Then R(G/UG )UH will be invariant under the(left) action of AH , and we may decomposeR(G/UG )UH into eigenspacesfor AH . Since the functions in R(G/UG )UH are by definition (left) in-


variant under UH , the (left) AH -eigenfunctions will in fact be (left) BH

eigenfunctions. In other words, they are highest weight vectors for H.Hence, the characters of AH acting on (the left of) R(G/UG )UH will allbe dominant with respect to BH , and we may write R(G/UG )UH as asum of (left) AH eigenspaces (R(G/UG )UH )χ for dominant characters χ

of H:

R(G/UG )UH =⊕

χ∈A+H

(R(G/UG )UH )χ . (2.3)

Since the spaces Vψ of decomposition (2.1) are (left) G-invariant, theyare a fortiori left H-invariant, so we have a decomposition ofR(G/UG )UH

into right AG -eigenspaces (R(G/UG )UH )ψ :

R(G/UG )UH =⊕

ψ∈A+G

R(G/UG )UH ∩ Vψ :=⊕

ψ∈A+G

R(G/UG )UH

ψ .

Combining this decomposition with the decomposition (2.3), we maywrite

R(G/UG )UH =⊕

ψ∈A+G , χ∈A+

H

(R(G/UG )UH

ψ )χ . (2.4)

To emphasize the key features of this algebra, we note the resultingconsequences of decomposition (2.4) in the following proposition.

Proposition 2.2

(a) The decomposition (2.4) is an (A+G × A+

H )-graded algebra decom-position of R(G/UG )UH .

(b) The subspaces (R(G/UG )UH

ψ )χ tell us the χ highest weight vectorsfor BH in the irreducible representation Vψ of G. Therefore, thedecomposition

R(G/UG )UH

ψ =⊕

χ∈A+H

(R(G/UG )UH

ψ )χ

tells us how Vψ decomposes as a H-module.

Thus, knowledge of R(G/UG )UH as a (A+G × A+

H )-graded algebra tellus how representations of G decompose when restricted to H, in otherwords, it describes the branching rule from G to H.We will call R(G/UG )UH the (G,H) branching algebra. When G H × H, and H is embedded diagonally in G, the branching algebradescribes the decomposition of tensor products of representations of H,and we then call it the tensor product algebra for H. More generally, we


would like to understand the (G,H) branching algebras for symmetricpairs (G,H).

In the context of regular functions on G/U , such branching algebrasare a little too abstract. We shall elaborate later on a more concreteconstruction of branching algebras, which allows substantial manipula-tion. We shall come back to the concrete construction at the end of §4,after some preliminaries in §3 and an example in §4.1.

3 Preliminaries and Notations

3.1 Parametrization of Representations

Let G be a classical reductive algebraic group over C: G = GLn (C) =GLn , the general linear group; or G = On (C) = On , the orthogonalgroup; or G = Sp2n (C) = Sp2n , the symplectic group. We shall explainour notations on irreducible representations of G using integer partitions.In each of these cases, we select a Borel subalgebra of the classical Liealgebra and coordinatize it, as is done in [11]. Consequently, all highestweights are parameterized in the standard way (see [11]).

A non-negative integer partition λ, with k parts, is an integer sequenceλ1 ≥ λ2 ≥ . . . ≥ λk > 0. We may sometimes refer to λ as a Young orFerrers diagram. We use the same notation for partitions as is done in[50]. For example, we write (λ) to denote the length (or depth) of apartition, i.e., (λ) = k for the above partition. Also let |λ| =

∑i λi be

the size of a partition and λ′ denote the transpose (or conjugate) of λ

(i.e., (λ′)i = |λj : λj ≥ i|).

GLn Representations: Given non-negative integers p, q and n suchthat n ≥ p + q and non-negative integer partitions λ+ and λ− withp and q parts respectively, let F

(λ+ ,λ−)(n) denote the irreducible rational

representation of GLn with highest weight given by the n-tuple:

(λ+ , λ−) =(λ+

1 , λ+2 , · · · , λ+

p , 0, · · · , 0,−λ−q , · · · ,−λ−

1

)︸︷︷︸n

If λ− = (0) then we will write Fλ+

(n) for F(λ+ ,λ−)(n) . Note that if λ+ = (0)

then(Fλ−

(n)

)∗is equivalent to F

(λ+ ,λ−)(n) . More generally,

(F

(λ+ ,λ−)(n)

)∗is

equivalent to F(λ−,λ+ )(n) .

On Representations: The complex orthogonal group has two con-


nected components. Because the group is disconnected we cannot indexirreducible representations by highest weights. There is however an ana-log of Schur-Weyl duality for the case of On in which each irreduciblerational representation is indexed uniquely by a non-negative integerpartition ν such that (ν′)1 + (ν′)2 ≤ n. That is, the sum of the first twocolumns of the Young diagram of ν is at most n. We will call such adiagram On -admissible (see [11] Chapter 10 for details). Let Eν

(n) denotethe irreducible representation of On indexed ν in this way.

An irreducible rational representation of SOn may be indexed by itshighest weight. In [11] Section 5.2.2, the irreducible representations ofOn are determined in terms of their restrictions to SOn (which is anormal subgroup having index 2). We note that if (ν) = n

2 , then therestriction of Eν

(n) to SOn is irreducible. If (ν) = n2 (n even), then

Eν(n) decomposes into exactly two irreducible representations of SOn .

See [11] Section 10.2.4 and 10.2.5 for the correspondence between thisparametrization and the above parametrization by partitions.

The determinant defines an (irreducible) one-dimensional representa-tion of On . This representation is indexed by the length n partitionζ = (1, 1, · · · , 1). An irreducible representation of On will remain ir-reducible when tensored by Eζ

(n) , but the resulting representation maybe inequivalent to the initial representation. We say that a pair of On -admissible partitions α and β are associate if Eα

(n) ⊗ Eζ(n)

∼= Eβ(n) . It

turns out that α and β are associate exactly when (α′)1 + (β′)1 = n and(α′)i = (β′)i for all i > 1. This relation is clearly symmetric, and isrelated to the structure of the underlying SOn -representations. Indeed,when restricted to SOn , Eα

(n)∼= Eβ

(n) if and only if α and β are eitherassociate or equal.

3.2 Multiplicity-Free Actions

Let G be a complex reductive algebraic group acting on a complex vec-tor space V . We say V is a multiplicity-free action if the algebra P(V )of polynomial functions on V is multiplicity free as a G module. Thecriterion of Servedio-Vinberg-Kimel′fel′d ([58, 62]) says that V is mul-tiplicity free if and only if a Borel subgroup B of G has a Zariski openorbit in V . In other words, B (and hence G) acts prehomogeneouslyon V (see [57]). A direct consequence is that B eigenfunctions in P(V )have a very simple structure. Let Qψ ∈ P(V ) be a B eigenfunction witheigencharacter ψ, normalized so that Qψ (v0) = 1 for some fixed v0 ina Zariski open B orbit in V . Then Qψ is completely determined by ψ:


For v = b−1v0 in the Zariski open B orbit,

Qψ (v) = Qψ (b−1v0) = ψ(b)Qψ (v0) = ψ(b), b ∈ B.

Qψ is then determined on all of V by continuity. Since B = AU , andU = (B,B) is the commutator subgroup of B, we can identify a characterof B with a character of A. Thus the B eigenfunctions are precisely theG highest weight vectors (with respect to B) in P(V ). Further

Qψ1 Qψ2 = Qψ1 ψ2

and so the set of A+(V ) = ψ ∈ A+ | Qψ = 0 forms a sub-semigroupof the cone A+ of dominant weights of A.

An element ψ(= 1) of a semigroup is primitive if it is not expressibleas a non-trivial product of two elements of the semigroup. The algebraP(V )U has unique factorization (see [23]). The eigenfunctions associatedto the primitive elements of A+(V ) are prime polynomials, and P(V )U

is the polynomial ring on these eigenfunctions. If ψ = ψ1ψ2 , then Qψ =Qψ1 Qψ2 . Thus, if ψ is not primitive, then the polynomial Qψ cannot beprime. An element

ψ = Πkj=1ψ

cj

j

has cj ’s uniquely determined, and hence the prime factorization

Qψ = Πkj=1Q

cj

ψj.

Consider a multiplicity-free action of G on an algebra W. In thegeneral situation, we would like to associate this algebra W with a sub-algebra of R(G/U). With this goal in mind, we introduce the followingnotion:

Definition 7 Let P =⊕

λ∈A+ Pλ denote an algebra graded by an abeliansemigroup A+ . If W ⊆ P is a subalgebra of P, then we say that W is atotal subalgebra of P if

W =⊕λ∈Z

Pλ

where Z is a sub-semigroup of A+ , which we will denote by A+(W) = Z.Note that W is graded by A+(W).

In what is to follow, we will usually have P = P(V ) (polynomialfunctions on a vector space V ) and A+ will denote the dominant chamber


of the character group of a maximal torus A of a reductive group G actingon V . In this situation, we introduce an A+ -filtration on W as follows:

W(ψ ) =⊕ϕ≤ψ

Wϕ (3.1)

where the ordering ≤ is the ordering on A+ given by (see [54])

ψ1 ≤ ψ2 if ψ−11 ψ2 is expressible as a product of

rational powers of positive roots.

Note that positive roots are weights of the adjoint representation of G onits Lie algebra g. We refer to the abelian group structure on the integralweights multiplicatively. Also, it will turn out that we only need positiveinteger powers of the positive roots.

Next consider the more specific situation where W which is a G-invariant and G-multiplicity-free subalgebra of a polynomial algebraP(V ). Suppose that WU has unique factorization. Then WU is a poly-nomial ring and A+(W) is a free sub-semigroup in A+ generated by thehighest weights corresponding to the non-zero graded components of W.Write the G decomposition as follows:

W =⊕

ψ∈A+ (W)

Wψ

noting that Wψ is an irreducible G module with highest weight ψ.If δ occurs with positive multiplicity in the tensor product decompo-

sition

Wϕ ⊗Wψ =⊕

δ

dim HomG (Wδ ,Wϕ ⊗Wψ ) Wδ ,

then δ ≤ ϕψ. From 3.1 we can see that if

Wη ⊂ W(ϕ) and Wγ ⊂ W(ψ ) , i.e., η ≤ ϕ and γ ≤ ψ,

then it follows that

Wη · Wγ →Wη ⊗Wγ ⊂ W(ηγ ) ⊂ W(ϕψ ) .

ThusW(ϕ) · W(ψ ) ⊂ W(ϕψ ) .

We have now an A+ -filtered algebraW =

⋃ψ∈A+ (W)

W(ψ ) ,

and this filtration is known as the dominance filtration [54].


With a filtered algebra, we can form its associated algebra which isA+ graded:

GrA+W =⊕

ψ∈A+ (W)

(GrA+W)ψ

where

(GrA+W)ψ = W(ψ )/

⊕ϕ<ψ

W(ϕ)

.

Theorem 3.1 Consider a multiplicity-free G-module W with a A+ -filtered algebra structure such that W is a unique factorization domain.Assume that the zero degree subspace of W is C. Then there is a canon-ical A+ -graded algebra injection:

GrA+ π : GrA+W → R(G/U).

Note that this result immediately follows from a more general theorem(see Theorem 5 of [54] and the Appendix to [61]). Moreover, both theassumption on the zero degree subspace and the unique factorization canbe removed. We thank the referee for providing both these observationsand the references. Below is a proof cast in our present context.

Proof of Theorem 3.1. In [16] it is shown that under the abovehypothesis, WU is a polynomial ring on a canonical set of generators.Now, WU is a A+ -graded algebra and therefore, there exists an injectiveA+ -graded algebra homomorphism obtained by sending each generatorof the domain to a (indeed any) highest weight vector of the same weightin the codomain:

α : WU → R(G/U)U .

Note that WU = GrA+ (WU ) = (GrA+W)U .There exists a unique G-module homomorphism

α : GrA+W → R(G/U)

such that the following diagram commutes:

α : WU → R(G/U)U

∩ ∩α : GrA+W → R(G/U)


We wish to show that α is an algebra homomorphism, i.e.,

(GrA+W)λ × (GrA+W)µ −→mW

(GrA+W)λ+µ

α ↓ α ↓ α ↓

R(G/U)λ × R(G/U)µ −→mR(G / U )

R(G/U)λ+µ

commutes.We have two maps:

fi : (GrA+W)λ ⊗ (GrA+W)µ → R(G/U)λ+µ , i = 1, 2,

defined by: f1(v ⊗ w) = mR(G/U )(α(v) ⊗ α(w)) and f2(v ⊗ w) =α(mW(v ⊗ w)).

Each of f1 and f2 is G-equivariant and,

dimβ∣∣ β : (GrA+W)λ ⊗ (GrA+W)µ → R(G/U)λ+µ

= 1

because the Cartan product has multiplicity one in the tensor productof two irreducible G-modules Vλ and Vµ (this is a well known fact thatis not difficult to prove, see for example [54]).

Therefore, there exists a constant C such that f1 = Cf2 . We knowthat α|WU = α is an algebra homomorphism. So for highest weightvectors vλ ∈ WU

λ and wµ ∈ WUµ :

f1(vλ ⊗ wµ) = α(vλ)α(wµ) = α(vλ )α(wµ)

= α(vλwµ) = α(vλwµ) = f2(vλ ⊗ wµ).

(Note that vλwµ is a highest weight vector.) Note that C = 1.

3.3 Dual Pairs and Duality Correspondence

The theory of dual pairs will feature prominently in this article. For thetreatment of all the branching algebras arising from classical symmetricpairs, we will need to understand dual pairs in three different settings.However, to minimize exposition, we restrict our discussion to just oneof the pairs. Details in this section including the more general cases canbe found in [11], [14] or [16].

In our context, the theory of dual pairs may be cast in a purelyalgebraic language. In this section, we will describe the dual pairs


(On, sp2m ). Let On to be the group of invertible n × n matrices, g

such that gJgt = J where J is the n× n matrix:

J =

0 · · · 0 1... · · · 1 0

0 1 · · ·...

1 0 · · · 0

.

Let Mn,m be the vector space of n×m complex matrices, and considerthe polynomial algebra P(Mn,m ) over Mn,m . The group On ×GLm actson P(Mn,m ) by (g, h) · f(x) = f(gtxh), where g ∈ On , h ∈ GLm andx ∈Mn,m . The derived actions of their Lie algebras act on P(Mn,m ) bypolynomial coefficient differential operators. Using the standard matri-ces entries as coordinates, we define the following differential operators:

∆ij :=n∑

s=1

∂2

∂xsi∂xn−s+1, j, r2

ij :=n∑

s=1

xsixn−s+1, j , and

Eij :=n∑

s=1

xsi∂

∂xsj.

We define three spaces:

sp(1,1)2m := Span

Eij + n

2 δi,j | i, j = 1, . . . ,m glm ,

sp(2,0)2m := Span

r2ij | 1 ≤ i ≤ j ≤ m

, and (3.1)

sp(0,2)2m := Span ∆ij | 1 ≤ i ≤ j ≤ m .

The direct sum, g := sp(2,0)2m ⊕ sp

(1,1)2m ⊕ sp

(0,2)2m , is preserved under the

usual operator bracket and is isomorphic, as a Lie algebra, to the rankm complex symplectic Lie algebra, sp2m . This presentation defines anaction of sp2m on P(Mn,m ).

Let S2Cm be the space of symmetric m by m matrices respectively. IfV is a vector space, we denote the symmetric algebra on V by S(V ). Fora set S, we shall denote by C[S] by the algebra generated by elementsin the set S.

Theorem 3.2 Invariants and Harmonics of On

(a) First Fundamental Theorem of Invariant Theory and Separation


of Variables. The invariants

Jn,m := P(Mn,m )On = C[r2ij ]

(∼= S(sp(2,0)

2m ) ∼= P(S2Cm ) if n ≥ m)

.

Further, if n ≥ 2m, we have separation of variables

P(Mn,m ) Hn,m ⊗ Jn,m .

where

Hn,m = f ∈ P(Mn,m ) | ∆ij f = 0 for all 1 ≤ i ≤ j ≤ m

denotes the On -harmonics in P(Mn,m ).(b) Multiplicity-Free Decomposition under On × sp2m . We have the

decomposition

P(Mn,m ) |On ×sp2 m=⊕

Eλ(n) ⊗ Eλ

(2m )

where the sum is over all partitions λ with length at most min(n,m),and such that (λ′)1 + (λ′)2 ≤ n. Furthermore, Eλ

(2m ) is an ir-reducible (infinite dimensional) highest weight representation ofsp2m such that as a representation of GLm ,

Eλ(2m ) = Jn,m · Fλ

(m ) for any n,m ≥ 0,∼= S(S2Cm )⊗ Fλ

(m ) provided n ≥ 2m.

(c) Multiplicity-Free Decomposition of Harmonics under On×sp(1,1)2m .

The On -harmonics Hn,m is invariant under the action of On ×sp

(1,1)2m . Here sp

(1,1)2m glm , and as an On ×GLm representation,

P(Mn,m )/I(J +n,m ) ∼= Hn,m =

⊕Eλ

(n) ⊗ Fλ(m ) ,

where the sum is over all partitions λ with length at most min(n,m)and such that (λ′)1 + (λ′)2 ≤ n. Here I(J +

n,m ) refers to the idealgenerated by the positive degree On invariants in P(Mn,m ).

4 Reciprocity Algebras

In this paper, we study branching algebras using classical invariant the-ory. The formulation of classical invariant theory in terms of dual pairs[14] allows one to realize branching algebras for classical symmetric pairsas concrete algebras of polynomials on vector spaces. Furthermore, whenrealized in this way, the branching algebras have a double interpreta-tion in which they solve two related branching problems simultaneously.


Classical invariant theory also provides a flexible means which allows aninductive approach to the computation of branching algebras, and makesevident natural connections between different branching algebras.

The easiest illustration of the above assertions is the realization of thetensor product algebra for GLn presented as follows. This example alsoillustrates the definition of a reciprocity algebra.

4.1 Illustration: Tensor Product Algebra for GLn

This first example is in [16], which we recall here as it is a model forthe other (more involved) constructions of branching algebras as totalsubalgebras (see Definition 3.1) of GLn tensor product algebras.

Consider the joint action of GLn ×GLm on the P(Mn,m ) by the rule

(g, h) · f(x) = f(gtxh), for g ∈ GLn, h ∈ GLm , x ∈ Mn,m .

For the corresponding action on polynomials, one has the GLn × GLm

multiplicity free decomposition (see [16])

P(Mn,m ) ⊕

λ

Fλ(n) ⊗ Fλ

(m ) , (4.1)

of the polynomials into irreducible GLn × GLm representations. Notethat the sum is over non-negative partitions λ with length at mostmin (n,m).

Let Um = UGLmdenote the upper triangular unipotent subgroup of

GLm . From decomposition (4.1), we can easily see that

P(Mn,m )Um (⊕

λ

Fλ(n) ⊗ Fλ

(m )

)Um

⊕

λ

Fλ(n) ⊗ (Fλ

(m ))Um . (4.2)

Since the spaces (Fλ(m ))

Um are one-dimensional, the sum in equation(4.2) consists of one copy of each Fλ

(n) . Just as in the discussion of §3.2,

the algebra is graded by A+m , where Am is the diagonal torus of GLm ,

and one sees from (4.2) that the graded components are the Fλ(n) .

By the arguments in §3.2, P(Mn,m )Um can thus be associated to agraded subalgebra in R(GLn/Un ), in particular, this is a total sub-algebra as in Definition 3.1. To study tensor products of represen-tations of GLn , we can take the direct sum of Mn,m and Mn, . Wethen have an action of GLn × GLm × GL on P(Mn,m ⊕Mn,). SinceP(Mn,m ⊕Mn,) P(Mn,m )⊗P (Mn,), we may deduce from (4.1) that

P(Mn,m ⊕Mn,)Um ×U P(Mn,m )Um ⊗ P(Mn,)U


⊕µ,ν

(Fµ(n) ⊗ Fν

(n))⊗((Fµ

(m ))Um ⊗ (Fν

())U

). (4.3)

Thus, this algebra is the sum of one copy of each tensor productsFµ

(n)⊗Fν(n) . Hence, if we take the Un -invariants, we will get a subalgebra

of the tensor product algebra for GLn . This results in the algebra

(P(Mn,m ⊕Mn,)Um ×U )Un P(Mn,m ⊕Mn,)Um ×U ×Un .

This shows that we can realize the tensor product algebra for GLn ,or more precisely, various total subalgebras of it, as algebras of poly-nomial functions on matrices, specifically as the algebras P(Mn,m ⊕Mn,)Um ×U ×Un .

However, the algebra P(Mn,m ⊕Mn,)Um ×U ×Un has a second inter-pretation, as a different branching algebra. We note that Mn,m⊕Mn, Mn,m+ . On this space we have the action of GLn × GLm+ , which isdescribed by the obvious adaptation of equation (4.1). The action ofGLn × GLm × GL arises by restriction of the action of GLm+ to thesubgroup GLm × GL embedded block diagonally in GLm+ . By (theobvious analog of) decomposition (4.2), we see that

P(Mn,m+)Un ⊕

λ

(Fλ(n))

Un ⊗ Fλ(m+) .

This algebra embeds as a subalgebra of R(GLm+/Um+), in particular,this is a total subalgebra as in Definition 3.1. If we then take the Um×U

invariants, we find that

(P(Mn,m+)Un )Um ×U ⊕

λ

(Fλ(n))

Un ⊗ (Fλ(m+))

Um ×U

is (a total subalgebra of) the (GLm+ , GLm ×GL) branching algebra.Thus, we have established the following result.

Theorem 4.1

(a) The algebra P(Mn,m+)Un ×Um ×U is isomorphic to a total sub-algebra of the (GLn × GLn,GLn ) branching algebra (a.k.a. theGLn tensor product algebra), and to a total subalgebra of the(GLm+ , GLm ×GL) branching algebra.

(b) In particular, the dimension of the ψλ × ψµ × ψν homogeneouscomponent for An × Am × A of P(Mn,m+)Un ×Um ×U recordssimultaneously

(i) the multiplicity of Fλ(n) in the tensor product Fµ

(n) ⊗ Fν(n), and


(ii) the multiplicity of Fµ(m ) ⊗ Fν

() in Fλ(m+),

for partitions µ, ν, λ such that (µ) ≤ min(n,m), (ν) ≤ min(n, )and (λ) ≤ min(n,m + ).

Thus, we can not only realize the GLn tensor product algebra con-cretely as an algebra of polynomials, we find that it appears simultane-ously in two guises, the second being as the branching algebra for thepair (GLm+ , GLm ×GL). We emphasize two features of this situation.

First, the pair (GLm+ , GLm × GL), as well as the pair (GLn ×GLn,GLn ), is a symmetric pair. Hence, both the interpretations ofP(Mn,m+)Un ×Um ×U are as branching algebras for symmetric pairs.

Second, the relationship between the two situations is captured bythe notion of “see-saw pair” of dual pairs [41]. Precisely, a contextfor understanding the decomposition law (4.1) is provided by observ-ing that GLn and GLm (or more correctly, slight modifications of theirLie algebras) are mutual centralizers inside the Lie algebra sp(Mn,m )(of the metaplectic group) of polynomial coefficient differential oper-ators of total degree two on Mn,m [14] [16]. We say that they de-fine a dual pair inside sp(Mn,m ). The decomposition (4.1) then ap-pears as the correspondence of representations associated to this dualpair [14]. Further, the pairs of groups (GLn,GLm+) = (G1 , G

′1) and

(GLn ×GLn,GLm ×GL) = (G2 , G′2) both define dual pairs inside the

Lie algebra sp(Mn,m+). We evidently have the relations

G1 = GLn ⊂ GLn ×GLn = G2 , (4.4)

and (hence)

G′1 = GLm+ ⊃ GLm ×GL = G′

2 . (4.5)

We refer to a pair of dual pairs related as in inclusions (4.4) and (4.5),a see-saw pair of dual pairs.

In these terms, we may think of the symmetric pairs (G2 , G1) and(G′

1 , G′2) as a “reciprocal pair” of symmetric pairs. If we do so, we see that

the algebra P(Mn,m+)Un ×Um ×U is describable as P(Mn,m+)UG 1 ×UG ′

2

– it has a description in terms of the see-saw pair, and in this descriptionthe two pairs of the see-saw, or alternatively, the two reciprocal sym-metric pairs, enter equivalently into the description of the algebra thatdescribes the branching law for both symmetric pairs. For this reason,we also call this algebra, which describes the branching law for bothsymmetric pairs, the reciprocity algebra of the pair of pairs.


It turns out that any branching algebra associated to a classical sym-metric pair, that is, a pair (G,H) in which G is a product of classi-cal groups, has an interpretation as a reciprocity algebra – an algebrathat describes a branching law for two reciprocal symmetric pairs si-multaneously. Sometimes, however, one of the branching laws involvesinfinite-dimensional representations.

4.2 Symmetric Pairs and Reciprocity Pairs

In the context of dual pairs, we would like to understand the (G,H)branching of irreducible representations of G to H, for symmetric pairs(G,H). Table I lists the symmetric pairs which we will cover in thispaper.

If G is a classical group over C, then G can be embedded as onemember of a dual pair in the symplectic group as described in [14]. Theresulting pairs of groups are (GLn,GLm ) or (On, Sp2m ), each insideSp2nm , and are called irreducible dual pairs. In general, a dual pair ofreductive groups in Sp2r is a product of such pairs.

Table I: Classical Symmetric Pairs

Description G H

Diagonal GLn × GLn GLn

Diagonal On × On On

Diagonal Sp2n × Sp2n Sp2n

Direct Sum GLn +m GLn × GLm

Direct Sum On +m On × Om

Direct Sum Sp2(n +m ) Sp2n × Sp2m

Polarization O2n GLn

Polarization Sp2n GLn

Bilinear Form GLn On

Bilinear Form GL2n Sp2n


Proposition 4.2 Let G be a classical group, or a product of two copiesof a classical group. Let G belong to a dual pair (G,G′) in a symplecticgroup Sp2m . Let H ⊂ G be a symmetric subgroup, and let H ′ be thecentralizer of H in Sp2m . Then (H,H ′) is also a dual pair in Sp2m ,and G′ is a symmetric subgroup inside H ′.

Proof: This can be shown by fairly easy case-by-case checking. Thebasic reason that (H,H ′) form a dual pair is that, for any classicalsymmetric pair (G,H), the restriction of the standard module of G, orits dual, to H is a sum of standard modules of H, or their duals [14]. Thisis very easy to check on a case-by-case basis. The see-saw relationshipof symmetric pairs organizes the 10 series of symmetric pairs as given inTable I into five pairs of series. These are shown in Table II.

Table II: Reciprocity Pairs

SymmetricPair(G, H) (H, h′) (G, g′)(GLn × GLn , GLn ) (GLn , glm + ) (GLn × GLn , glm ⊕ gll )(On × On , On ) (On , sp2(m + ) ) (On × On , sp2m ⊕ sp2 l )(Sp2n × Sp2n , Sp2n ) (Sp2n , so2(m + ) ) (Sp2n × Sp2n , so2m ⊕ so2 )(GLn +m , GLn × GLm ) (GLn × GLm , gl ⊕ gl ) (GLn +m , gl )(On +m , On × Om ) (On × Om , sp2 ⊕ sp2 ) (On +m , sp2 )(Sp2(n +m ) , Sp2n × Sp2m ) (Sp2n × Sp2m , so2 ⊕ so2 ) (Sp2(n +m ) , so2 )(O2n , GLn ) (GLn , gl2m ) (O2n , sp2m )(Sp2n , GLn ) (GLn , gl2m ) (Sp2n , so2m )(GLn , On ) (On , sp2m ) (GLn , glm )(GL2n , Sp2n ) (Sp2n , so2m ) (GL2n , glm )

Remark: Table II also amounts to another point of view on the struc-ture on which [15] is based.

As alluded to in §2, we need a more concrete realization of branchingalgebras. With this goal in mind, we shall introduce the general defini-tion of a reciprocity algebra through the following sequence of steps:

Step 1 Consider a symmetric pair (G,H).Step 2 Use the theory of dual pairs to construct a multiplicity free G×

K variety V, for a group K associated to a dual pair involving G.Analogues to the Theory of Spherical Harmonics (see Theorem3.3) allow us to consider a dual pair (G, g′), which has the Liealgebra of K as g′(1,1) (similar to the space sp

(1,1)2m in (3.1)). We


note that g′ forms a family of Lie algebras, and each choice of g′

determines the type of G irreducible representations involved.Step 3 Consider the coordinate ring of V, which we denote by C[V].

Note that C[V] is either polynomial algebra or a quotient of apolynomial algebra, depending on which dual pair we are con-sidering. If UK is a maximal unipotent subgroup of K, thenC[V]UK is a partial model of G, in other words, a collection ofirreducible representations of G appearing once and only oncein C[V]UK .

Step 4 Taking UH covariants, the algebra C[V]UK ×UH will be our can-didate. We will abuse our terminology and still call it a "branch-ing algebra". This is because C[V]UK ×UH sits in R(G/UG )UH

as a total subalgebra (see Definition 3.1). We hasten to add, aspointed out at the end of Step 2, that we have a family of totalsubalgebras in R(G/UG )UH . Further, each total subalgebra re-lates two branching phenomena, and thus we call it a reciprocityalgebra.

In the following table we provide the ingredients for the special casesas well as the stability range for the classical branching formula involvingLittlewood-Richardson coefficients (see [21]).

Table III: Stability Range

Sym. Pair, (G,H) K Rep. of G×K Stability Range

(GLk ,Ok ) GLp ×GLq Mk,p ⊕M∗k,q k ≥ 2(p + q)

(GL2k , Sp2k ) GLp ×GLq M2k,p ⊕M∗2k,q k ≥ p + q

(O2k ,GLk ) GL2n M2k,2n k ≥ 2n

(Sp2k ,GLk ) GL2n M2k,2n k ≥ 2n

(GLk+ , GLk ×GL) GLp ×GLq Mk+,p ⊕M∗k+,q min(k, l) ≥ p + q

(Ok+ , Ok ×O) GLn Mk+,n min(k, l) ≥ 2n

(Sp2k+2 , Sp2k × Sp2) GLn M2(k+),n min(k, l) ≥ n

(GLk ×GLk ,GLk )GLp ×GLq×GLr ×GLs

Mk,p ⊕Mk,q⊕M∗

k,r ⊕M∗k,s

k ≥ p + q + r + s

(Ok ×Ok ,Ok ) GLn ×GLm Mk,n+m k ≥ 2(n + m)(Sp2k × Sp2k , Sp2k ) GLn ×GLm M2k,n+m k ≥ (n + m)

With the above general construction in mind, we begin with two ofthe reciprocity algebras in the next two sections. We have chosen theexamples more to illustrate the subtleties and the general framework.


5 Branching from GLn to On

Consider the problem of restricting irreducible representations of GLn

to the orthogonal group On . We consider the symmetric see-saw pair(GLn,On ) and (Sp2m ,GLm ). As in the discussion of §4.1, we can realize(a total subalgebra of) the coordinate ring of the flag manifold GLn/Un

as the algebra of Um -invariants on P(Mn,m ). If we then look at the UOn-

invariants in this algebra, then we will have (a certain total subalgebraof) the (GLn,On ) branching algebra. Thus, we are interested in thealgebra

P(Mn,m )UO n ×Um .

We note that, in analogy with the situation of §4.1, this is the algebraof invariants for the unipotent subgroups of the smaller member of eachsymmetric pair.

Let us investigate what this algebra appears to be if we first takeinvariants with respect to UOn

. We have a decomposition of P(Mn,m )as a joint On × sp2m -module (see Theorem 3.3(b)):

P(Mn,m ) ⊕

µ

Eµ(n) ⊗ Eµ

(2m ) . (5.1)

Recall that the sum runs through the set of all non-negative integer par-titions µ such that (µ) ≤ min(n,m) and (µ′)1 + (µ′)2 ≤ n. Here Eµ

(n)denotes the irreducible On representation parameterized by µ. Recallfrom (4.1), the multiplicity free GLn ×GLm decomposition P(Mn,m ) ⊕

µ Fµ(n) ⊗ Fµ

(m ) . The module Eµ(n) is generated by the GLn high-

est weight vector in Fµ(n) . Further, Eµ

(2m ) is an irreducible infinite-dimensional representation of sp2m with lowest glm -type Fµ

(m ) .

Theorem 5.1 Assume n > 2m.

(a) The algebra P(Mn,m )UO n ×Um is isomorphic to a total subalgebraof the (GLn,On ) branching algebra, and to a total subalgebra ofthe (sp2m ,GLm ) branching algebra.

(b) In particular, the dimension of the ϕµ ×ψλ homogeneous compo-nent for AOn

×Am of P(Mn,m )UO n ×Um records simultaneously

(i) the multiplicity of Eµ(n) in the representation Fλ

(n), and

(ii) the multiplicity of Fλ(m ) in Eµ

(2m ).

for partitions µ, λ such that (µ) ≤ m, and (λ) ≤ m.


Proof. Taking the UOn-invariants for the decomposition (5.1), we find

that

P(Mn,m )UO n ⊕

µ

(Eµ(n))

UO n ⊗ Eµ(2m ) , (5.2)

where the sum is over partitions µ such that (µ) ≤ min(n,m) and(µ′)1 + (µ′)2 ≤ n. Note that the stability condition n > 2m guaranteesthe latter inequality. The space (Eµ

(n))UO n is the space of highest weight

vectors for (Eµ(n))

UO n . We would like to say that it is one-dimensional,so that P(Mn,m )UO n would consist of one copy of each of the irreduciblerepresentations Eµ

(2m ) . But, owing to the disconnectedness of On , thisis not quite true, and when it is true, the highest weight may not com-pletely determine Eµ

(n) .However, if n > 2m, then (Eµ

(n))UO n is one-dimensional, and does

single out Eµ(n) among the representations which appear in the sum (5.1).

Hence, let us make this restriction for the present discussion. Taking theUm invariants in the sum (5.2), we find that

(P(Mn,m )UO n )Um ⊕

µ

(Eµ(n))

UO n ⊗ (Eµ(2m ))

Um . (5.3)

Note that the sum is over all partitions µ such that (µ) ≤ m (sincen > 2m). The space (Eµ

(2m ))Um describes how the representation Eµ

(2m )of sp2m decomposes as a glm module, or equivalently, as a GLm -module.In other words, (Eµ

(2m ))Um describes the branching rule from sp2m to glm

for the module Eµ(2m ) .

We know (thanks to our restriction to n > 2m) that the space (Eµ(n))

UO n

is one-dimensional. Let ϕµ be the AOnweight of (Eµ

(n))UO n . Thus, ϕµ is

the restriction to the diagonal maximal torus AOnof the character ψµ of

the group An of diagonal n×n matrices. Our assumption further impliesthat ϕµ determines Eµ

(n) . Therefore, for a given dominant Am weight ψλ ,corresponding to the partition λ, where (λ) ≤ m, the ψλ -eigenspace in(Eµ

(2m ))Um tells us the multiplicity of Fλ

(m ) in the restriction of Eµ(2m ) to

glm . This is the same as the dimension of the joint (ϕµ×ψλ)-eigenspacein

(P(Mn,m )UO n )Um P(Mn,m )UO n ×Um (P(Mn,m )Um )UO n .

But we have already seen that this eigenspace describes the multiplic-ity of Eµ

(n) in Fλ(n) . Thus, again the AOn

× Am homogeneous compo-nents of P(Mn,m )UO n ×Um have a simultaneous interpretation, one for a


branching law associated to each of the two symmetric pairs composingthe symmetric see-saw pair.

In this case, one of the branching laws involves infinite-dimensionalrepresentations. However, they are highest weight representations, whichare the most tractable of infinite-dimensional representations, from analgebraic point of view.

6 Tensor Product Algebra for On

Using the symmetric see-saw pair((On ×On,On ), (Sp2(m+) , Sp2m × Sp2)

),

we can construct (total subalgebras of) the tensor product algebra forOn . To prepare for this, we should explicate the decomposition (5.1)further.

Let us recall the basic setup as in §3.3. Recall that Jn,m = P(Mn,m )On

is the algebra of On -invariant polynomials. Theorem 3.3(a) implies thatJn,m is a quotient of S(sp(2,0)

2m ), the symmetric algebra on sp(2,0)2m .

The natural mapping

Hn,m → P(Mn,m )/I(J +n,m )

is a linear On × GLm -module isomorphism. Further, the On × GLm

structure of Hn,m is as follows (see Theorem 3.3(c)):

Hn,m ⊕

µ

Eµ(n) ⊗ Fµ

(m ) .

Here µ ranges over the same diagrams as in (5.1).From Theorem 3.3(b),

Eµ(2m ) Fµ

(m ) · Jn,m S(sp(2,0)2m ) · Fµ

(m ) , (6.1)

and it follows that

Eµ(2m )/(sp(2,0)

2m · Eµ(2m )) Fµ

(m ) .

In other words, we can detect the sp2m isomorphism class of the moduleEµ

(2m ) by the GLm isomorphism class of the quotient Eµ(2m )/(sp(2,0)

2m ·Eµ

(2m )). Also, if W ⊂ P(Mn,m ) is any sp2m -invariant subspace, then

W/(sp(2,0)2m ·W ) W ∩Hn,m ,

and this subspace also reveals the sp2m isomorphism type of W .


We can use the above to find a total subalgebra of the tensor productalgebra of On . One consequence of the above discussion is that(

P(Mn,m )/I(J +n,m )

)Um ⊕

µ

Eµ(n) ⊗ (Fµ

(m ))Um

consists of one copy of each irreducible representation Eµ(n) .

If we repeat the above discussion for Mn, , and combine the results,we find that(

P(Mn,m )/I(J +n,m )

)Um ⊗(P(Mn,)/I(J +

n,))U

⊕µ,ν

(Eµ

(n) ⊗ Eν(n)

)⊗((Fµ

(m ))Um ⊗ (Fν

())U

)(6.2)

is a direct sum of one copy of each possible tensor product of an Eµ(n)

with an Eν(n) . At this point, we make the assumption that n > 2(m+ ),

as in this range the On constituents of decomposition are irreduciblewhen restricted to the connected component of the identity in On . Ifwe now take the UOn

-invariants in equation (6.2), we will have (a totalsubalgebra of) the tensor product algebra of On :(

(P(Mn,m )/I(J +n,m ))Um ⊗ (P(Mn,)/I(J +

n,))U

)UO n

⊕µ,ν

(Eµ

(n) ⊗ Eν(n)

)UO n ⊗((Fµ

(m ))Um ⊗ (Fν

())U

).

We can describe this algebra in another way. Begin with the observa-tion that P(Mn,m )⊗ P(Mn,) P(Mn,m+), and

P(Mn,m )/I(J +n,m )⊗ P(Mn,)/I(J +

n,) P(Mn,m+)/I(J +n,m ⊕ J +

n,).

Thus


n,))U

(P(Mn,m+)/I(J +n,m ⊕ J +

n,))Um ×U ,

and taking UOninvariants of the above, we get(


n,))U

)UO n

((P(Mn,m+)/I(J +

n,m ⊕ J +n,))

Um ×U

)UO n

((P(Mn,m+)/I(J +

n,m ⊕ J +n,))

UO n

)Um ×U

.


Theorem 6.1 Given positive integers n, m and with n > 2(m + ) wehave:

(a) The algebra((P(Mn,m )/I(J +

n,m ))Um ⊗ (P(Mn,)/I(J +n,))

U

)UO n

is isomorphic to a total subalgebra of the (On×On,On ) branchingalgebra (a.k.a. the On tensor product algebra), and to a totalsubalgebra of the (sp2(m+) , sp2m ⊕ sp2) branching algebra.

(b) Specifically, the dimension of the (ϕλ × ψµ × ψν )-eigenspace for

AOn×Am×A of

((P(Mn,m+)/I(J +

n,m ⊕ J +n,))

UO n

)Um ×U

recordssimultaneously

(i) the multiplicity of Eλ(n) in Eµ

(n) ⊗Eν(n), as well as

(ii) the multiplicity of Eµ(2m )⊗ Eν

(2) in the restriction of Eλ(2(m+)).

Here the partitions µ, ν, λ satisfy the following conditions:(µ) ≤ min(n,m), (ν) ≤ min(n, ), and (λ) ≤ min(n,m + ).

Proof. Let us now compute the ring expressed in this way. FromTheorem 3.3(b), we know that

P(Mn,m )UO n (⊕

µ

Eµ(n) ⊗ Eµ

(2m )

)UO n

⊕

µ

(Eµ(n))

UO n ⊗ Eµ(2m ) .

Note that within the range n > 2(m + ) we have dim(Eµ(n))

UO n = 1since the On -representations Eµ

(n) remain irreducible when restricted toSOn .

Now repeat this with m replaced by m + :

P(Mn,m+)UO n (⊕

µ

Eµ(n) ⊗ Eµ

(2(m+))

)UO n

⊕

µ

(Eµ(n))

UO n ⊗ Eµ(2(m+)) .

Hence

(P(Mn,m+)/I(J +n,m ⊕ J +

n,))UO n

((⊕

λ

Eλ(n) ⊗ Eλ

(2(m+))

)/I(J +

n,m ⊕ J +n,)

)UO n

⊕

λ

(Eλ(n))

UO n ⊗(Eλ

(2(m+))/(sp(2,0)2m ⊕ sp

(2,0)2 ) · Eλ

(2(m+))

).


From this we finally get((P(Mn,m+)/I(J +

n,m ⊕ J +n,))

UO n

)Um ×U

⊕

λ

(Eλ(n))

UO n ⊗(Eλ

(2(m+))/(sp(2,0)2m ⊕ sp

(2,0)2 ) · Eλ

(2(m+))

)Um ×U

.

From the discussion following equation (6.1), we see that the factor(Eλ

(2(m+))/(sp(2,0)2m ⊕ sp

(2,0)2 ) · Eλ

(2(m+))

)Um ×U

tells us the sp2m ⊕ sp2 decomposition of Eλ(2(m+)) .

Hence, again the algebra has a double interpretation, one in terms ofdecomposing tensor products of On representations, and one in terms ofbranching from sp2(m+) to sp2m ⊕ sp2 (although the second branchinglaw involves infinite-dimensional representations).

7 The Stable Range and Relations Between ReciprocityAlgebras

Let us summarize our discussions this far. Given any classical symmetricpair, we can embed it in a (family of) see-saw symmetric pair(s). Doingthis, we find that (a total subalgebra of) the branching algebra for thepair can equally well be interpreted as the branching algebra for a dualfamily of representations of the dual symmetric pair. The representa-tions of the dual symmetric pair will frequently be infinite dimensional,but they are always highest weight modules.

An immediate consequence of this isomorphism of algebras is the iso-morphisms of intertwining spaces and hence equality of multiplicities,which we have collectively described as reciprocity laws. These reci-procity laws are of the same nature as Frobenius Reciprocity for inducedrepresentations of groups.

From §4.2, we see that the see-saw symmetric pairs actually come intwo parameter families. If one of the pairs involves many more variablesthan the other, then certain features of the discussions above becomesimpler.

Take the results of Theorem 4.1 as an illustration: Let n, m and

denote positive integers. Now suppose that λ, µ and ν are partitionssuch that the length of λ (resp. µ, resp. ν) is at most min(n,m + )(resp. min(n,m), resp. min(n, )). Then the Littlewood-Richardson


coefficient

cλµν = dim HomGLn

(Fλ(n) , F

µ(n) ⊗ Fν

(n))

while at the same time,

cλµν = dim HomGLm ×GL

(Fµ(m ) ⊗ Fν

() , Fλ(m+)).

Thus, for fixed λ, µ and ν, we have two distinct interpretations of theLittlewood-Richardson coefficients for sufficiently large n, m and .

Consider another example: branching from GLn to On . If we letthese groups act on P(Mn,m ), we get the see-saw pairs (On, sp2m ) and(GLn,GLm ). The branching coefficients dµ

λ from GLn to On can bedescribed as follows:

Fλ(n) |On

=∑

µ

dµλ Eµ

(n)

wheredµ

λ = dim HomOn(Eµ

(n) , Fλ(n))

= dim HomGLm

(Fλ

(m ) , Fµ(m ) ⊗ S(sp(2,0)

2m ))

= dim HomGLm

(Fλ

(m ) , Fµ(m ) ⊗ S(S2Cm )

)is independent of n, if n ≥ m, and only depends on the diagrams λ

and µ. This allows one to create a theory of “stable characters” for On .Similar considerations apply to GLn and Sp2n and this idea has beenactively pursued by [39], amongst others.

These are all instances of stability laws. The well-known one-stepbranching from GLn to GLn−1 is another instance. More precisely, thedominant weights of GLn (resp. GLn−1) may be indexed by partitionsas in Section 3.1. Relative to this parametrization, the only requirementis that n be sufficiently large when compared to the lengths of the par-titions. In other words, a single partition indexes a highest weight of arepresentation of GLn for all sufficiently large n. Thus, this branchingcan be described entirely by diagrams, with no mention of the size n, ifn is large. Iteration of this branching also shows that when n is large,the weight multiplicities of dominant weights of an irreducible GLn rep-resentation are independent of n, in a similar sense. See [1] for the otherclassical groups, which don’t share this stability property.

In the last two sections that follow, we will illustrate the simplifica-tions that occur in the stable range, highlighting certain specific see-sawpairs. In all these cases, we show that the branching algebras associated


to symmetric pairs can all be described by use of suitable branchingalgebras associated to the general linear groups. Thus, if we can havecontrol of the solution in the general linear group case, we will have somecontrol of the other classical groups. The other non-trivial examples willbe important extensions of this work, and we will see them in furtherpapers, for example, [21], [22], [17], [18] and [19].

8 Stability for Branching from GLn to On

We begin with a detailed discussion of the case of

(GLn,On ) and (sp2m ,GLm ).

Here we have already encountered the stable range, without the name.It is when n > 2m. Several things happen in the stable range:

(a) The representations Eµ(n) of the orthogonal group remain irre-

ducible when restricted to the special orthogonal group SOn , andfurthermore, no two of them are equivalent.

(b) Recall the algebra Jn,m of On -invariant polynomials on Mn,m gen-erated by the quadratic invariants, which is the abelian subalgebrasp

(2,0)2m of sp2m . In the stable range (in fact it holds true whenever

n ≥ m), the natural surjective homomorphism

S(sp(2,0)2m ) → Jn,m

is an isomorphism. See Theorem 3.3(a).(c) In the stable range, the multiplication map

Hn,m ⊗ Jn,m Hn,m ⊗ S(sp(2,0)2m ) → P(Mn,m )

is also an isomorphism of On×GLm -modules. See Theorem 3.3(a)and 3.3(b).

Of course, the subspace Hn,m of harmonic polynomials is not an al-gebra – it is not closed under multiplication. This is quite clear, sinceHn,m contains all the linear functions, which generate the whole poly-nomial ring. However, to form the reciprocity algebra associated to thesymmetric see-saw pairs (GLn,On ) and (sp2m ,GLm ), we need to takethe UOn

-invariants. Thus, our reciprocity algebra is a subalgebra of

(Hn,m ⊗ S(sp(2,0)2m ))UO n = HUO n

n,m ⊗ S(sp(2,0)2m )

(⊕

µ

(Eµ(n))

UO n ⊗ Fµ(m )

)⊗ S(sp(2,0)

2m ). (8.1)


Theorem 8.1 When n > 2m, the space HUO nn,m is a subalgebra of P(Mn,m ).

Hence, the algebra P(Mn,m )UO n is isomorphic to a tensor product

P(Mn,m )UO n HUO nn,m ⊗ S(sp(2,0)

2m )

of the algebras HUO nn,m and S(sp(2,0)

2m ). Furthermore, the algebra HUO nn,m

is isomorphic (as a representation) to the subalgebra R+(GLm /Um ) ofR(GLm /Um ) defined by the polynomial representations.

Proof. Note thatHUO nn,m can be identified with a subalgebraR+(GLm /Um )

ofR(GLm /Um ) defined by the polynomial representations, from our dis-cussion in §3.2. Consider the space of polynomials belonging to the sumin the last expression of equation (8.1). Letxjk | j = 1, . . . , n, k = 1, . . . , m be the standard matrix entries onMn,m . In order to make the unipotent group UOn

of On maximallycompatible with (in fact, contained in) the unipotent subgroup Un ofGLn , we choose the inner product on Cn as in Section 3.3. By thischoice, joint On ×GLm harmonic highest weight vectors are monomialsin the determinants

δj = det

x11 x12 . . . x1j

x21 x22 . . . x1j

......

......

xj1 xj2 . . . xjj

for j = 1, . . . , m.

From this, we can see that the space∑

µ(Eµ(n))

UO n ⊗Fµ(m ) is spanned by

the monomials in the determinants

det

x1,b1 x1,b2 . . . x1,bj

x2,b1 x2,b2 . . . x2,bj

......

......

xj,b1 xj,b2 . . . xj,bj

(8.2)

as b1 , b2 , b3 , . . . , bj ranges over all j-tuples of integers from 1 to m.Indeed, the span of such monomials is clearly invariant under glm , andconsists of highest weight vectors for On . Finally, we see that thesemonomials will all be harmonic, because the partial Laplacians spanningsp

(0,2)2m have the form

∆ab =n∑

j=1

∂2

∂xj,a∂xn+1−j,b.

Since every term of ∆ab involves differentiating with respect to a vari-able xjk with j > n/2, and the determinants (8.2) do not depend on


these variables, we see that they will be annihilated by the ∆ab , whichmeans that they are harmonic. This shows that HUO n

n,m is a subalgebraof P(Mn,m ).

We have thus completed the proof of the theorem.

We can use the description in Theorem 8.1 of P(Mn,m )UO n to relatethe branching algebra P(Mn,m )UO n ×Um to the tensor product algebra forGLm . As a GLm -module, the space sp

(2,0)2m is isomorphic to S2Cm , the

space of symmetric m×m matrices. It is well known that the symmetricalgebra S(S2Cm ) is multiplicity-free as a representation of GLm , anddecomposes into a sum of one copy of each polynomial representationcorresponding to a diagram with rows of even length (or a partition ofeven parts):

S(S2Cm ) ⊕

ν

F 2ν(m ) .

(Note that this result is in several places in the literature. See [11] and[16] for example.)

As a GLm -module, S(S2Cm ) could be embedded inR(GLm /Um ), butthe algebra structures on these two algebras are quite different.

Using the dominance filtration (see §3.2), we have a canonical A+ -algebra filtration on S(S2Cm ). If we form the associated graded algebra,then Theorem 3.2 says that it will be isomorphic to the subalgebra ofR(GLm /Um ) spanned by the representations attached to diagrams witheven length rows.

Let us denote the associated graded algebra of S(S2Cm ) by

GrA+mS(S2Cm ).

Let us denote the subalgebra of R(GLm /Um ) spanned by the represen-tations attached to diagrams with even length rows by R+2(GLm /Um ).

We can filter the tensor product HUO nn,m ⊗ S(sp(2,0)

2m ) by means of thefiltration on S(sp(2,0)

2m ). The associated graded algebra will then beHUO n

n,m ⊗GrA+mS(sp(2,0)

2m ). This discussion has indicated that the followingresult holds.

Theorem 8.2 When n > 2m, the associated graded algebra of P(Mn,m )UO n

with respect to the dominance filtration on the factor Jn,m is isomor-phic to the tensor product of the graded subalgebras R+(GLm /Um ) andR+2(GLm /Um ) of R(GLm /Um ):

GrA+m

(P(Mn,m )UO n ) R+(GLm /Um )⊗R+2(GLm /Um ).


Of course, GrA+m

(P(Mn,m )UO n ) is isomorphic as a GLm -module toP(Mn,m )UO n in an obvious way, by construction. Also GrA+

m(P(Mn,m )UO n )

inherits the A+On

grading from P(Mn,m )UO n – it becomes identified withthe A+

m grading on the first factor R+(GLm /Um ) in the tensor productof Theorem 8.2. On the other hand, the second factor is also A+

m -gradedin the obvious way, since it is the factor which defines the associatedgraded. When we take the Um invariants, we get another grading byA+

m , associated to the Am action on the Um invariants. This triplyA+

m -graded algebra is evidently a total subalgebra of the tensor productalgebra of GLm .On the other hand, we could take the Um invariants inside P(Mn,m )UO n ,

and then pass to the associated graded. It is not hard to convince oneselfthat these two processes commute with each other. Hence, we finallyhave:

Corollary 8.3 When n > 2m, the associated graded algebra of Um

invariants in P(Mn,m )UO n ,

GrA+m

((P(Mn,m )UO n

)Um)(

GrA+m

(P(Mn,m )UO n ))Um

(R+(GLm /Um )⊗R+2(GLm /Um )

)Um

is a triply-graded total subalgebra of the tensor product algebra of GLm .The restrictions on the gradings which define GrA+

m

((P(Mn,m )UO n

)Um)

are:

(a) the weight on the first factor of (R(GLm /Um )⊗R(GLm /Um ))Um

should correspond to a partition (i.e., it should be a polynomialweight), and

(b) the weight on the second factor should correspond to a partitionwith even parts.

Remark: The content of Corollary 8.3 in terms of multiplicities is theLittlewood Restriction Formula [8], [21]; see formula (2.4.1), [28]; see(5.7) with (4.19), [39]; see Theorem 1.5.3 and 2.3.1, [47] and [48]. Withthis result it is possible to compute a basis of the reciprocity algebra for(GLn,On ) using [21]; see [18].


9 Tensor Products for On

According to Theorem 6.1, we can compute tensor products for theorthogonal group via the algebra((

P(Mn,m )/I(J +n,m )

)Um ⊗(P(Mn,)/I(J +

n,))U

)UO n

.

Here the stable range is n > 2(m + ). Then we have

P(Mn,m+) Hn,m+ ⊗ S(sp(2,0)2(m+)).

Furthermore,

S(sp(2,0)2(m+)) = S(sp(2,0)

2m ⊕ sp(2,0)2 ⊕ (Cm ⊗ C))

S(sp(2,0)2m )⊗ S(sp(2,0)

2 )⊗ S(Cm ⊗ C)

Since Jn,m S(sp(2,0)2m ) and Jn, S(sp(2,0)

2 ), we see that

P(Mn,m )/I(J +n,m )⊗P(Mn,)/I(J +

n,) P(Mn,m⊕Mn,)/I(J +n,m⊕J +

n,)

Hn,m+ ⊗ S(sp(2,0)2(m+))/I(sp(2,0)

2m ⊕ sp(2,0)2 ) (9.1)

Hn,m+ ⊗ S(Cm ⊗ C).

Thus, using equation (9.1), we see that(P(Mn,m )/I(J +

n,m )⊗ P(Mn,)/I(J +n,)

)UO n

(Hn,m+ ⊗ S(Cm ⊗ C)

)UO n HUO n

n,m+ ⊗ S(Cm ⊗ C)

(⊕

λ

Eλ(n) ⊗ Fλ

(m+)

)UO n

⊗ S(Cm ⊗ C)

(⊕

λ

(Eλ(n))

UO n ⊗ Fλ(m+)

)⊗ S(Cm ⊗ C)

(⊕

λ

(Fλ(n))

Un ⊗ Fλ(m+)

)⊗ S(Cm ⊗ C).

Note that Fλ(n) is the GLn representation generated by the highest weight

of the On representation Eλ(n) and both (Fλ

(n))Un and (Eλ

(n))On are one

dimensional.


Hence, finally we get((P(Mn,m )/I(J +

n,m ))Um ⊗

(P(Mn,)/I(J +

n,))U

)UO n

.

((P(Mn,m )/I(J +

n,m )⊗ P(Mn,)/I(J +n,)

)UO n

)Um ×U

((⊕

λ

(Fλ(n))

Un ⊗ Fλ(m+)

)⊗ S(Cm ⊗ C)

)Um ×U

. (9.2)

We can interpret this algebra in term of tensor product algebras forgeneral linear groups. Now consider the (polynomial) tensor productalgebras

(R+(GLk/Uk )⊗R+(GLk/Uk ))Uk ⊕λ,µ

(Fλ

(k) ⊗ Fµ(k)

)Uk

for k = n,m and . If we form the tensor product of these, we get

(R+(GLn/Un )⊗R+(GLn/Un ))Un ⊗(R+(GLm /Um )⊗R+(GLm /Um ))Um

⊗ (R+(GL/U)⊗R+(GL/U))U

⊕

α,β ,δ,λ,µ,ν

(Fα

(n) ⊗ Fβ(n)

)Un

⊗(Fδ

(m ) ⊗ Fλ(m )

)Um

⊗(Fµ

() ⊗ Fν()

)U

Let us denote this algebra by TTn,m, . The algebra TTn,m, is (A+n )3 ×

(A+m )3 × (A+

)3-graded. If we require that λ = α, or that µ = β, orthat ν = δ, then we obtain total subalgebras of TTn,m, . If δ = α, wewill denote it by ∆1,3TTn,m, , and so forth. The subalgebra obtained byrequiring that all three diagonal conditions occur at once will be denotedby using all three ∆’s. Thus we will write

∆1,3∆2,5∆4,6TTn,m,

=∑α,β ,δ

(Fα

(n) ⊗ Fβ(n)

)Un

⊗(Fα

(m ) ⊗ Fδ(m )

)Um

⊗(Fβ

() ⊗ Fδ()

)U

It takes a similar argument by expanding (9.2) (as in §8) to see that∆1,3∆2,5∆4,6TTn,m, and((

P(Mn,m )/I(J +n,m )⊗ P(Mn,)/I(J +

n,))UO n

)Um ×U


are isomorphic as multigraded vector spaces. They may not be isomor-phic as algebras, because((

P(Mn,m )/I(J +n,m )⊗ P(Mn,)/I(J +

n,))UO n

)Um ×U

is not graded, while we see that ∆1,3∆2,5∆4,6TTn,m, is. However, if wepass to the associated graded of S(Cm ⊗ C), then the two algebras dobecome isomorphic. We record this fact.

Theorem 9.1 Assume the stable range n > 2(m + ). We have thefollowing isomorphisms of (A+

n )3 × (A+m )3 × (A+

)3-graded algebras:

Gr(A+n )3 ×(A+

m )3 ×(A+ )3

(((P(Mn,m )/I(J +

n,m ))Um ⊗

(P(Mn,)/I(J +

n,))U

)UO n

)

∆1,3∆2,5∆4,6TTn,m, .

Remark: The content of Theorem 9.1 in terms of multiplicities can befound in [21]; see formula (2.1.2), [32]; see Theorem 4.1 and [52].

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11Character formulae from

Hermann Weyl to the presentJens Carsten JantzenMathematics Institute

Aarhus Universitet

Introduction

In 1926 Hermann Weyl published a paper that contains his character for-mula for irreducible finite dimensional complex representations of com-plex and real semi-simple Lie groups and their Lie algebras. It can alsobe interpreted as a character formula for connected compact groups andfor semi-simple algebraic groups in characteristic 0. (Here I am usingmodern terminology; when Weyl wrote his paper, terms like “Lie groups”were not yet in use.)

When we look at Weyl’s character formula as a statement for Liealgebras, then it is a theorem on purely algebraic objects. However,Weyl used analytic methods to prove it. Not surprisingly, people lookedfor algebraic proofs. These attempts were finally successful and led alsoto useful reformulations of Weyl’s formula. This development will bedescribed in the first section of this survey.

The other topic to be discussed will be the search for analogues toWeyl’s formula in more general cases. To start with, a finite dimen-sional complex semi-simple Lie algebra has an abundance of irreduciblerepresentations that are infinite dimensional. It was natural to look forcharacter formulae for at least some families of representations sharingfeatures of the finite dimensional ones — for example those generatedby a highest weight vector.

Furthermore, it was also natural to go beyond finite dimensional com-plex semi-simple Lie algebras. There are several algebraic objects thatshare many structural features with these Lie algebras and that havesimilar representation theories. Let me mention those that we look athere:

In the 1950s it was proved that semi-simple algebraic groups over an

232

Character formulae from Hermann Weyl to the present 233

algebraically closed field of prime characteristic are classified in the sameway as semi-simple algebraic groups over C and also that the classifi-cation of simple modules by their highest weights generalises from Cto prime characteristic. In the late 1960s Kac–Moody algebras wereconstructed as generalisations of finite dimensional semi-simple Lie al-gebras, and in the 1980s quantum groups were introduced as deforma-tions of universal enveloping algebras. Also for these new objects simplemodules with a highest weight play an important role.

In all three cases the quest for character formulae has been a centralactivity over the years. It turned out that there were strong connectionsbetween the different set-ups. And in all three cases the strongest resultscannot be proved with algebraic methods alone (at least so far).

I have to admit that a quite different story could be told under thetitle that I have given here. Weyl’s 1926 paper was also the startingpoint for harmonic analysis on semi-simple Lie groups. A survey of thisline of development —looking, for example, at the monumental workof Harish-Chandra including his character formula for discrete seriesrepresentations — would certainly require as much space as what I havedone here. Given the limited time I had for my lecture in Bielefeld Ihad to restrict myself and thus decided to select the topics closest to myown work and area of expertise.

1 Semi-simple complex Lie algebras

Weyl’s character formula can be regarded as a statement about repre-sentations of Lie algebras or about representations of groups (Lie groupsor algebraic groups). In either case it requires a certain amount of no-tation. We first develop the Lie algebra version. For unexplained termswe refer to [Hu1].

1.1. Let g be a finite dimensional semi-simple complex Lie algebraand h a Cartan subalgebra of g. For any g–module V and any λ ∈ h∗

we denote by

Vλ = v ∈ V | h v = λ(h) v for all h ∈ h

the weight space of V for the weight λ. The set of all λ ∈ h∗ with Vλ = 0is called the set of weights of V .

The set of non-zero weights of g (considered as a g–module under theadjoint action) is called the root system of g with respect to h; this setwill be denoted by R. One can choose a base α1 , α2 , . . . , αr for the

234 Jens Carsten Jantzen

root system R. This base determines a partial ordering on h∗ such thatλ ≤ µ if and only if there exist non-negative integers m1 ,m2 , . . . ,mr

such that µ− λ =∑r

i=1 miαi .Set R+ = α ∈ R | α > 0 . Then R is the disjoint union of R+

and −R+ . Let n+ be the direct sum of all gα with α ∈ R+ , let n−

be the direct sum of all gα with α ∈ −R+ . Then g has the triangulardecomposition g = n− ⊕ h⊕ n+ .

For each α ∈ R there exists a unique hα ∈ [gα , g−α ] ⊂ h such thatα(hα ) = 2. Then the reflection sα : h∗ → h∗ given by sα (µ) = µ−µ(hα )α

permutes the root system R. The subgroup of GL(h∗) generated by allsα with α ∈ R is called the Weyl group of g with respect to h; it will bedenoted by W .

We call

X := λ ∈ h∗ | λ(hα ) ∈ Z for all α ∈ R

the set of integral weights. It is a free abelian group of rank equal todim h. The Weyl group W permutes X; under this action each elementof X is conjugate to exactly one element in

X+ := λ ∈ X | λ(hα ) ≥ 0 for all α ∈ R+ .

Elements in X+ are called dominant integral weights.1.2. If V is a finite dimensional g–module, then all weights of V are

integral and V is the direct sum of its weight spaces: V =⊕

λ∈X Vλ .The formal character of V is then defined as an element in the groupring Z[X]:

ch V =∑λ∈X

dim Vλ e(λ)

where (e(λ) | λ ∈ X) is the standard basis of Z[X]. This character isinvariant under the action of the Weyl group W given by w · e(λ) =e(w(λ)) for all w ∈W and λ ∈ X.

If V is a simple finite dimensional g–module, then the set of weightsof V has a unique maximal weight, called the highest weight of V . Thishighest weight is dominant; conversely each dominant integral weightis the highest weight of exactly one (up to isomorphism) simple finitedimensional g–module. In other words, mapping V to its highest weightinduces a bijection from the set of isomorphism classes of simple finitedimensional g–modules onto the set X+ . For each λ ∈ X+ let V (λ)denote a simple finite dimensional g–module with highest weight λ.

Set ρ equal to half the sum of all positive roots, i.e., of all elements


in R+ . It turns out that ρ ∈ X, in fact that ρ(hαi) = 1 for all simple

roots αi .Now Weyl’s character formula states that

ch V (λ) =∑

w∈W det(w) e(w(λ + ρ))∑w∈W det(w) e(w(ρ))

. (1)

More precisely one should say that here the denominator is non-zero anddivides the numerator in the integral domain Z[X]; so the quotient is awell defined element in Z[X]. In many cases it is useful to write Weyl’sformula in the form

ch V (λ) =∑

w∈W det(w) e(w(λ + ρ)− ρ)∑w∈W det(w) e(w(ρ)− ρ)

, (2)

i.e., to multiply the numerator and the denominator by e(−ρ).From the character formula one deduces Weyl’s dimension formula:

dim V (λ) =∏

α∈R+

(λ + ρ)(hα )ρ(hα )

. (3)

There is a ring homomorphism ϕ : Z[X] → Z mapping each e(µ) to 1.One has then ϕ(ch V ) = dimV for any finite dimensional g–module V .Unfortunately a direct application of ϕ to (1) leads to a fraction ofthe form 0/0. Therefore a more complicated argument is needed thatinvolves the denominator formula∑

w∈W

det(w) e(w ρ− ρ) =∏

α∈R+

(1− e(−α)), (4)

see [W3], page 386, line 7.1.3. Let me now state how to regard Weyl’s character formula as a

result for algebraic groups. (See [Bo1], [Hu2] or [Sp3] for unexplainedterms and background.)

Consider a connected semi-simple algebraic group G over C withLie G = g. Then we can find a maximal torus H in G with Lie H = h.Here “maximal torus” is to be interpreted in the algebraic group senseas a direct product of multiplicative groups C×.

Let X(H) denote the group of all characters of H, i.e., of all homo-morphisms H → C× of algebraic groups. Taking the tangent map atthe identity is an injective group homomorphism d : X(H) → h∗. Theimage dX(H) is a subgroup of finite index in the lattice X of all integralweights. We have dX(H) = X if G is simply connected; for arbitrary G

the image contains with finite index the subgroup∑

α∈R Zα generatedby all roots.


If V is a finite dimensional G–module (so by definition the represen-tation G → GL(V ) is a homomorphism of algebraic groups), then V

becomes a g–module when we take the tangent map at the identity ofG → GL(V ). In the decomposition V =

⊕µ∈X Vµ into weight spaces

for h, only µ ∈ dX(H) can contribute non-zero summands. Any h ∈ H

acts on any Vd ν with ν ∈ X(H) as multiplication by ν(h). Given g ∈ G

let χV (g) denote the trace of g acting on V . We get then for all h ∈ H

χV (h) =∑

ν∈X (H )

ν(h) dim Vd ν .

An arbitrary g ∈ G has a Jordan decomposition g = gsgu into a a unipo-tent factor gu and a semisimple factor gs . One gets then χV (g) = χV (gs)since the representation G → GL(V ) preserves the Jordan decomposi-tion. Furthermore, gs is conjugate in G to some hg ∈ H and we getχV (g) = χV (gs) = χV (hg ). This shows that the formal character ch V

determines the character χV . (One can avoid the use of the Jordan de-composition and observe that the semisimple elements in G are densein G and that χV is continuous.)

If V is a simple G–module, then also the derived g–module is simple,hence isomorphic to some V (λ) with λ ∈ X+ ∩ dX(H). One can showconversely that each V (λ) with λ ∈ X+ ∩ dX(H) lifts uniquely to aG–module. Therefore these V (λ) are a system of representatives for theisomorphism classes of simple G–modules. So Weyl’s character formuladetermines the characters of all simple G–modules.

We can regard G also as a complex Lie group. The simple G–modulesabove yield then also the irreducible holomorphic finite dimensional rep-resentations of G. Furthermore, if GR is a real form of G, then weget also the irreducible complex representations of the real Lie groupGR (R). (These representations correspond to irreducible complex rep-resentations of the Lie algebra gR = Lie GR , hence to simple modulesover gR ⊗R C g.)

1.4. Weyl proved his character formula in a series of three papers[W1], [W2], [W3] from 1925/26 where he first deals with special lin-ear groups, then with symplectic and special orthogonal groups1 beforefinally dealing with the general case.

His main tool is the integration over a compact real form of G. Theproof uses that the orthogonality relations for irreducible characters,first proved by Frobenius and Schur for finite groups, extend to compactgroups. (This had already been observed by Schur.) Another important

1 then called Komplexgrupppen and Drehungsgruppen in German


ingredient in the proof is the connection between the representations ofthe group and its Lie algebra; one exploits the fact that a formal char-acter ch V is a Z–linear combination of sums of the form

∑ν∈W µ e(ν)

with µ ∈ X+ . For a version of this proof in modern terminology see[BtD], chap. VI, §1. See also Slodowy’s survey [Sl] discussing the use ofthe integration technique by Weyl and in earlier work.

Weyl’s results confirm the character formulae found by Schur in [S1](with completely different methods) for the irreducible polynomial re-presentations of general linear groups as well as Schur’s results in [S2]for the compact real orthogonal groups. When comparing with Schur’sresults, Weyl underlines that he (Weyl) can deal with all orthogonalgroups (real and complex), not only with the compact ones as Schurdid.

Side results of Weyl’s three papers are the introduction of Weyl groupsand the proof of the complete reducibility of finite dimensional repre-sentations of semi-simple complex Lie algebras.

1.5. As stated in the preceding subsection Weyl used methods fromanalysis to prove his character formula which clearly is a result on alge-braic objects. The earlier work by Schur showed that a purely algebraicapproach existed for special linear groups, while Brauer achieved thisfor orthogonal groups in [Br1]. But an algebraic proof working for allsemi-simple complex Lie algebras was first found by Freudenthal in hispaper [F] from 1954.

In order to state Freudenthal’s crucial result we need one extra bit ofnotation. The Killing form on g is non-degenerate on h and thus inducesan isomorphism h

∼−→ h∗ of vector spaces. We use this isomorphism totransport the Killing form over to h∗. We get thus a W–invariant bilinearform ( , ) on h∗ that is positive definite on

∑µ∈X Rµ.

Now Freudenthal’s formula states for all λ ∈ X+ and µ ∈ X that

((λ + ρ, λ + ρ)− (µ + ρ, µ + ρ)) dim V (λ)µ

= 2∑

α∈R+

∞∑i=1

dim V (λ)µ+iα (µ + iα, α). (1)

This formula allows an inductive calculation of all dim V (λ)µ startingwith the general fact that dim V (λ)λ = 1. Then one has to know that(µ + ρ, µ + ρ) < (λ + ρ, λ + ρ) for all µ ∈ X with V (λ)µ = 0 and µ = λ.Note that the sum on the right hand side has only finitely many non-zeroterms because for any α ∈ R+ there are only finitely many i ∈ N withµ + iα ≤ λ.


The proof of (1) involves a reduction to the case g = sl(2,C) anda comparison of two different calculations of the action of the Casimirelement for g on V (λ).

Once one has (1) one can deduce Weyl’s character formula using someclever tricks. However, today the importance of Freudenthal’s formulacomes from another direction: It yields the fastest known algorithm forthe calculation of the dimensions of all V (λ)µ . It is therefore used incomputer programs (such as the package LiE) for this purpose. Regret-tably analogous formulae are missing for the characters of more generalmodules that we shall discuss later on.

1.6. Another important version of the character formula was found byKostant in his paper [Ko] from 1959. It involves the Kostant partitionfunction P that associates to any µ ∈ X the number of R+–tuples(nα )α∈R+ of non-negative integers such that µ =

∑α∈R+ nαα. So P (µ)

is non-zero if and only if µ ∈∑

α∈R+ Nα.Now Kostant’s formula states for all λ ∈ X+ and µ ∈ X that

dim V (λ)µ =∑

w∈W

det(w)P (w(λ + ρ)− (µ + ρ)). (1)

This equation can be easily deduced from the denominator formula1.2(4). In fact, using 1.2(4) one checks that (1) and 1.2(2) are equivalent,cf. [Hu1], 24.3.

Actually, Kostant used a more complicated approach to deduce hisformula from Weyl’s one. But soon Cartier (see [Cr]) and Steinbergdiscovered the simpler argument.

Kostant’s formula is not as useful for practical calculations as Freuden-thal’s formula because of the large sum over all Weyl group elements andbecause of the many cancellations in the sum. Note that for dominant µ

all summands (µ+ iα, α) in 1.5(1) are positive. So there are no cancella-tions in Freudenthal’s formula in this case, and the invariance of ch V (λ)under W shows that it suffices to compute dim V (λ)µ for dominant µ.

1.7. One gets a better insight into Kostant’s formula if one works withcertain infinite dimensional g–modules. For any Lie algebra a denoteby U(a) its universal enveloping algebra. For each µ ∈ h∗ let Cµ denotethe one dimensional (h ⊕ n+)–module on which any h ∈ h acts as µ(h)and any x ∈ n+ acts as 0. The induced g–module

M(µ) := U(g)⊗U (h⊕n+ ) Cµ (1)

is now usually called the Verma module with highest weight µ.The map u → u ⊗ 1 is an isomorphism U(n−) ∼−→ M(µ) of vector


spaces. It takes a weight vector of weight ν in U(n−) to a weight vector ofweight ν+µ in M(µ). Now a look at a PBW basis for U(n−) constructedwith root vectors in n− yields that M(µ) is the direct sum of its weightspaces and that

ch M(µ) =∑ν∈h∗

P (µ− ν) e(ν). (2)

For this to make sense we have to extend the definition of the formalcharacter as in 1.2 to infinite dimensional g–modules that have finitedimensional weight spaces and are the direct sums of these weight spaces;these general formal characters live then in a completion of Z[h∗] andwill be written as infinite sums of all e(ν) with ν ∈ h∗. These moregeneral characters need not be invariant under the action of the Weylgroup.

Now Kostant’s formula can be rewritten as

ch V (λ) =∑

w∈W

det(w) chM(w•λ) for all λ ∈ X+ (3)

where we use the “dot notation”

w•µ = w(µ + ρ)− ρ for all w ∈W and µ ∈ h∗. (4)

Actually, something stronger is true: There is for each λ ∈ X+ an exactsequence of g–modules (called the BGG resolution)

0 →M(w0•λ) → · · · (5)

→⊕

l(w )=2

M(w•λ) →⊕

l(w )=1

M(w•λ)→ M(λ)→ V (λ) → 0.

Here l(w) denotes for any w ∈ W the minimal length of an expressionw = sαi 1

sαi 2. . . sαi r

as product of reflections with respect to simpleroots αj . Furthermore w0 is the unique element in W of maximal length,equal to dim n+ ; it is also characterised as the unique element in W withw0(R+) = −R+ . So each term in (5) is the direct sum of all M(w•λ) withw ∈ W and l(w) = i for some i with 0 ≤ i ≤ dim n+ — except for theterms V (λ) and 0, of course. Since each reflection has determinant −1,we get det(w) = (−1)l(w ) for all w ∈ W . Therefore (3) follows easilyfrom (5) taking formal characters.

The surjection from M(λ) onto V (λ) appears already in the work ofChevalley [Ch] and Harish-Chandra [HC] who use this map to give analgebraic construction of the finite dimensional modules for g that doesnot proceed case-by-case. From Harish-Chandra’s work one also can


deduce that the kernel of this surjection is the image of the direct sumof all M(w•λ) with l(w) = 1. The existence of the whole exact sequencewas then proved by Bernštein, Gel′fand, and Gel′fand in [BGG2]. Thisarticle appeared in 1975, but had been available for a few years beforethat date. Their proof uses detailed knowledge about homomorphismsbetween Verma modules and about the Bruhat–Chevalley ordering onthe Weyl group; it involves a bit of homological algebra.

Note that each M(µ) is isomorphic to U(n−) as an n−–module. There-fore (5) is a free resolution of V (λ) considered as a U(n−)–module and wecan compute the Ext-groups Ext•n−(V (λ),C) as the cohomology groupsof the complex Homn−(M•,C) where C is the trivial one dimensionaln−–module and where Mi is the direct sum of all M(w•λ) with l(w) = i.Note that this is a complex of h–modules.

For any n−–module V we have a natural identification

Homn−(V,C) ∼−→ (V/n−V )∗.

For any µ ∈ h∗ the h–module M(µ)/n−M(µ) is isomorphic to Cµ . Itfollows that the i–th term in the complex Homn−(M•,C) is isomorphicto the direct sum of all C−w•λ with l(w) = i. The w•λ with w ∈ W

are pairwise distinct; therefore all maps in the complex are 0. It followsthat

Extin−(V (λ),C)

⊕l(w )=i

C−w•λ =⊕

l(w )=i

C−w (λ+ρ)+ρ . (6)

Exchanging R+ with −R+ this implies

Extin+ (V (λ),C)

⊕l(w )=i

C−w (w 0 λ−ρ)−ρ =⊕

l(w )=i

Cw•(−w 0 λ) . (7)

(One has to replace ρ by −ρ and the highest weight λ by the lowestweight w0 λ.) The isomorphisms Exti

n+ (V (λ),C) Extin+ (C, V (λ)∗) =

Hi(n+ , V (λ)∗) yield now

Hi(n+ , V (λ)) =⊕

l(w )=i

Cw•λ (8)

since V (λ)∗ V (−w0 λ). We get in particular Bott’s theorem on thedimensions of these cohomology groups.

1.8. The fastest way to prove Kostant’s (or Weyl’s) character formulaalgebraically is not to use the resolution 1.7(5). Instead one uses someelementary properties of the Verma modules. Any M(µ) with µ ∈ h∗

has a unique simple factor module that we denote by L(µ). For λ ∈ X+


we have L(λ) = V (λ). Using Harish-Chandra’s description of the centreof U(g) one shows that each M(µ) has finite length with all compositionfactors of the form L(w•µ) with w ∈W and w•µ ≤ µ; the simple moduleL(µ) occurs with multiplicity 1 in a composition series. This implies thatthere exist integers aµ,w ≥ 0 with

ch M(µ) = ch L(µ) +∑

w•µ<µ

aµ,w ch L(w•µ). (1)

From this we get integers bµ,w ∈ Z with

ch L(µ) =∑

w•µ≤µ

bµ,w ch M(w•µ) (2)

such that bµ,1 = 1.Now in order to prove Kostant’s formula in the form 1.7(3) we just

have to show for all λ ∈ X+ that bλ,w = det w for all w ∈ W . This,however, is an easy consequence of the W–invariance of ch V (λ).

This proof was discovered by Bernštein, Gel′fand, and Gel′fand intheir 1971 paper [BGG1]. It can be found in [Hu1], 24.2 with a simpli-fication in an appendix.

We have here looked at algebraic proofs of Weyl’s formula that usethe point of view of Lie algebras. There are also algebraic proofs thatuse the point of view of algebraic groups. Let me mention papers bySpringer [Sp1], Demazure [De], Donkin [Do], and Andersen [A1].

1.9. Another way of looking at Weyl’s character formula is Littel-mann’s path model developed in his papers [Li1] and [Li2] from 1994/95.Here a path is a piecewise linear, continuous map π from the interval[0, 1]Q := t ∈ Q | 0 ≤ t ≤ 1 to XQ :=

∑ν∈X Qν ⊂ h∗ such that

π(0) = 0. One identifies paths π and π′ if there exists a piecewise lin-ear, surjective, non-decreasing, continuous map ϕ : [0, 1]Q → [0, 1]Qwith π′ = π ϕ. These identifications ensure that the concatenationof paths is associative. Let P denote the set of all paths modulo theseidentifications.

In the path model one associates to each dominant weight λ ∈ X+ asubset Pλ of P such that

ch V (λ) =∑

π∈Pλ

e(π(1)). (1)

In other words, for each µ there are dim V (λ)µ paths in Pλ ending at µ.According to [Li2] there are several possible choices for Pλ . In [Li1] a

special choice was made, the set of Lakshmibai–Seshadri paths (short:


L–S paths), cf. Section 4 in [Li2]. One of these paths is πλ given byπλ(t) = tλ for all t, 0 ≤ t ≤ 1. The other L–S paths arise then by ap-plying Littelmann’s root operators: To each simple root αi he associatestwo root operators, ei and fi , that map P to P ∪ 0, see Section 1in [Li2]. If fi(π) = 0 for some π ∈ P, then fi(π)(1) = π(1) − αi ; ifei(π) = 0, then ei(π)(1) = π(1) + αi . Now Pλ consists of all paths ofthe form fi1 f12 · · · fis

(πλ ).One can associate to Pλ an oriented coloured graph whose vertices

are the elements of Pλ and which has an arrow πi−→ π′ if and only

if π′ = fi(π). This graph turns out to be isomorphic to the crystalgraph associated to V (λ) by Kashiwara using the corresponding quan-tum group, cf. [Jo2], 6.4.27.

One application of the path model is a formula for the decompositionof any tensor product V (λ) ⊗ V (λ′) with λ, λ′ ∈ X into a direct sumof simple submodules. This formula has the great advantage that themultiplicities are sums of non-negative terms, not alternating sums as inthe formulae by Brauer in [Br2] (see also note 22 to chap. VII in [W4])or Steinberg in [St].

1.10. We now leave the consideration of finite dimensional g–modulesand turn to character formulae for more general, but similar objects. Afirst example involves the in general infinite dimensional simple highestweight modules L(µ) from 1.8. The characters of these modules are ofinterest not only in themselves, but also because they determine char-acters for simple Harish-Chandra modules for the complex Lie group G

considered as a Lie group over the real numbers, cf. [BG], [E], or [Jo1].For the sake of simplicity let us concentrate on modules of the form

L(w•λ) with λ ∈ X+ and w ∈W . There are by 1.8(2) integers bw,x with

ch L(w•λ) =∑

x•λ≤w•λ

bw,x ch M(x•λ). (1)

Here the coefficients bw,x are actually independent of λ; this was provedin [Ja1].

In their 1979 paper [KL1] Kazhdan and Lusztig came up with a conjec-ture on the coefficients bw,x that was then proved by Brylinski and Kashi-wara in [BK] and independently by Beilinson and Bernstein, see [BB].The Kazhdan–Lusztig conjecture (now a theorem) states that

bw,x = det(xw)Pxw 0 ,ww 0 (1) (2)

where Px,w is the Kazhdan–Lusztig polynomial attached to w and x.These polynomials are constructed using the Hecke algebra of W . The


construction leads to a recursion formula that determines the polyno-mials uniquely and allows (in principle) their calculation. For anotherdescription of these polynomials and some variations of them, see [So3].

Soon after stating their conjecture in [KL1], Kazhdan and Lusztig gavein [KL2] a geometric interpretation of their polynomials. It involves theflag variety B := G/B where B is the Borel subgroup of G with Lie B =h⊕n+ . For each w ∈ W let Bw denote the Bruhat cell Bw = BwB/B. Onthe closure Bw one considers now the intersection cohomology complexIC•(Bw ). This is an object in the derived category Db

cs(Bw ) of sheaveson Bw with constructible cohomology sheaves; its restriction to the opensubset Bw is the constant sheaf C put into degree −l(w) = −dimBw .Now the result in [KL2] says for any x ∈W with Bx ⊂ Bw that

Px,w (t2) = tl(w )∑

i

dimHi(IC•(Bw ))x ti . (3)

Here Hi(IC•(Bw )) denotes the i-th cohomology sheaf of the complexIC•(Bw ) and the index x indicates that we take the stalk at any pointof Bx ; the dimension in (3) does not depend on the choice of this pointsince the restriction of Hi(IC•(Bw )) to Bx turns out to be a constantsheaf. An alternative proof of (3) due to MacPherson can be foundin [Sp2].

In case Bx ⊂ Bw one usually does not define Px,w . However, we setPx,w = 0 in this case in order to simplify the formulation of (2).

1.11. Let me describe the main ingredients of the proof of 1.10(2).For more details see [Sp2].

Given 1.10(3) it appears reasonable to find some connection betweencomplexes of sheaves on B and modules of the form L(w•λ) and M(w•λ).This can be done using the sheaf DB of (algebraic) differential operatorson B.

The action of G on B = G/B by left translation induces an algebrahomomorphism from the universal enveloping algebra U(g) to the alge-bra DB(B) of global sections of DB. This homomorphism turns out to besurjective; its kernel is the ideal I0U(g) generated by the annihilator I0

of the trivial one dimensional g–module C = L(0) in the centre of U(g).Setting U 0 = U(g)/I0U(g) we get an isomorphism U 0 ∼−→ DB(B), hencean equivalence between U 0–modules and DB(B)–modules . AllL(w•0) and M(w•0) with w ∈W are U 0–modules.

Next one shows that DB(B)–modules and DB–modules are equi-valent categories: One maps any DB–module M to the DB(B)–moduleM(B) of its global sections, while any DB(B)–module M is sent to the


DB–module DB⊗DB(B)M . Combining this with the earlier result one hasnow an equivalence between U 0–modules and DB–modules . Forany w ∈ W let Lw denote the DB–module corresponding to L(ww0•0)and Mw the one corresponding to M(ww0•0). (Note the factor w0 .)

All Lw and Mw belong to a special class of DB–modules, the holo-nomic modules with regular singularities. On this class the functorRHomDB(OB, ) takes values in Db

cs(B); in fact it induces an equivalenceof derived categories. (This is the Riemann–Hilbert correspondence.)

One shows that RHomDB(OB,Lw ) is the intersection cohomologycomplex IC•(Bw ) extended by 0 outside Bw and shifted in degree bydimB, whereas RHomDB(OB,Mw ) is the constant sheaf CBw

on Bw

extended by 0 outside Bw and put in degree dimB − l(w).Let us write [M ] for the class of an object M in a suitable Grothendieck

group. Now 1.10(1) can be rewritten [L(w•λ)] =∑

x bw,x [M(x•λ)]. Theequivalence between U 0–modules and DB–modules yields then[Lw ] =

∑x bww 0 ,xw 0 [Mx ]. And since RHomDB(OB, ) is an equivalence

on the subcategory involved, one gets

(−1)dim B[IC•(Bw )] =∑

x

bww 0 ,xw 0 (−1)dim B−l(x) [CBx]. (1)

(Note that a shift in degree by d in Dbcs(B) corresponds to a multiplica-

tion by (−1)d in the Grothendieck group.)For any z ∈ B the map taking any X in Db

cs(B) to∑

i(−1)i dimHi(X)z

factors through the Grothendieck group of Dbcs(B). Applying it to (1)

with z ∈ Bx for some x ∈ W we get∑i

(−1)i dimHi(IC•(Bw ))x = (−1)l(x)bww 0 ,xw 0 . (2)

By 1.10(3) the left hand side in (2) is equal to (−1)l(w ) Px,w ((−1)2) =det(w)Px,w (1). This now yields 1.10(2) since w2

0 = 1.Finally, I have to admit to some cheating above. In order to get the

Riemann–Hilbert correspondence one has to consider B as an analyticmanifold. In the functor RHomDB(OB, ) we have to replace OB by thesheaf Oan

B of analytic functions on B and similarly DB by the sheaf DanB

of differential operators with analytic coefficients. If M is a DB–module,then this functor has to be applied to Man := Oan

B ⊗OB M.1.12. In 1.10/11 we have restricted to L(µ) with µ ∈ W •λ for some

λ ∈ X+ . An arbitrary µ ∈ X can be written in the form µ = w•λ withλ ∈ −ρ+X+ ; here λ is uniquely determined by µ, but w is not except forλ ∈ X+ . However, we can make w unique by requiring that l(w) is maxi-


mal among all l(w′) with µ = w′•λ. If we do this, then 1.10(2) extends tothis case and we get ch L(µ) =

∑x det(wx)Pxw 0 ,ww 0 (1) ch L(x•λ); this

follows from [Ja1].For arbitrary µ ∈ h∗ one has to consider Rµ := α ∈ R | µ(hα ) ∈ Z

and Wµ := w ∈ W | wµ − µ ∈ ZR . One checks that Rµ is a rootsystem with Weyl group Wµ . Given µ one can find w ∈ Wµ and λ ∈ h∗

with µ = w•λ and (λ + ρ)(hα ) ≥ 0 for all α ∈ Rµ ∩ R+ . If here(λ + ρ)(hα ) > 0 for all α ∈ Rµ ∩R+ , then one gets

ch L(µ) =∑

x

det(wx)Pλxwλ ,wwλ

(1) ch L(x•λ) (1)

where we sum over x ∈ Wµ = Wλ , with wλ the longest element in Wλ ,and where Pλ

xwλ ,wwλdenotes a Kazhdan–Lusztig polynomial for the re-

flection group Wλ . If (λ + ρ)(hα ) = 0 for some α ∈ Rµ ∩ R+ , then (1)extends provided one chooses w of maximal length (taken in Wλ ) withµ = w•λ. One gets (1) from the equivalence of categories constructedin [So1] though this is not explicitly stated there.

2 Kac–Moody algebras

We now turn to Kac–Moody algebras. Their structure is very similar tothat of the finite dimensional semi-simple Lie algebras from Section 1.We take the notation from that classical case and add a sub- or super-script “KM” to denote the corresponding object in the Kac–Moody case.For more background on Kac–Moody algebras one may consult [Kc2],[MP], [Wa], or [Ca].

2.1. A Kac–Moody algebra is a Lie algebra over C associated to ageneralised Cartan matrix, i.e., an I×I–matrix A = (aij ) for some finiteindex set I such that all entries in A belong to Z, such that aii = 2 forall i and aij ≤ 0 whenever i = j; in addition one requires that aij = 0if and only if aji = 0. This last condition is always satisfied when A issymmetrisable, which means that there exist integers di > 0 such thatdiaij = djaji for all i and j. A Kac–Moody algebra associated to asymmetrisable generalised Cartan matrix is called symmetrisable.

The Kac–Moody algebra gKM associated to A has a triangular decom-position gKM = n

−KM ⊕hKM ⊕n

+KM . The subalgebra hKM is commutative

and finite dimensional. As in the classical case gKM is the direct sumof weight spaces for hKM under the adjoint action. The 0–weight spaceis hKM itself; the non-zero weights form the root system RKM ⊂ h∗

KM


of gKM . It is the disjoint union of R+KM and −R+

KM where R+KM consists

of the α ∈ RKM such that the root space gKM ,α is contained in n+KM .

Then n−KM is the direct sum of all gKM ,α with α ∈ −R+

KM . Any rootspace gKM ,α is finite dimensional; its dimension may well be greaterthan 1.

There are linearly independent elements (hi)i∈I in hKM and linearlyindependent elements (αi)i∈I in h∗

KM such that αi(hj ) = aji for all i

and j. Usually these families are not bases for their respective spaces;the only exception is the case det A = 0. Each αi is a positive root;one has R+

KM ⊂∑

i∈I Nαi . As in 1.1 one uses the αi to define a partialordering ≤ on h∗

KM .For each i ∈ I let si denote the reflection on h∗

KM defined by si(λ) =λ−λ(hi)αi . The group WKM generated by all si is called the Weyl groupof gKM . The root system RKM is stable under WKM . A root is calledreal if it belongs to some WKMαi with i ∈ I; all other roots are calledimaginary.

If A is the Cartan matrix of the Lie algebra g from 1.1, then gKM isisomorphic to g. If A is not the Cartan matrix of some finite dimensionalcomplex semi-simple Lie algebra, then gKM is infinite dimensional, infact RKM and WKM are infinite.

If gKM is symmetrisable, then there exists a non-degenerate bilinearform ( , ) on h∗

KM invariant under WKM . It satisfies (αi, αi) = 0 andλ(hi) = 2 (λ, αi)/(αi, αi) for all i and all λ ∈ h∗

KM . When we use sucha form, we shall assume that it is normalised so that each (αi, αi) is apositive real number. In this case a root α is imaginary if and only if(α, α) ≤ 0.

2.2. As in 1.7 any µ ∈ h∗KM defines a one dimensional module Cµ

over hKM ⊕ n+KM and then an induced Verma module

MKM(µ) := U(gKM)⊗U (hK M ⊕n+K M ) Cµ . (1)

It is a direct sum of finite dimensional weight spaces. So one can de-fine its character ch MKM(µ) in a suitable completion of the group ringZ[h∗

KM] and gets

ch MKM(µ) = e(µ)∏

α∈R+K M

(1− e(−α))−mα (2)

where mα = dim gKM ,α = dim gKM ,−α .Any MKM(µ) has a unique simple quotient that we denote by LKM(µ).

In contrast to the classical case Verma modules usually have infinitelength. However, given ν ∈ h∗

KM one can find a submodule N of MKM(µ)


such that ν is not a weight of N and such that MKM(µ)/N has finitelength with all composition factors of the form LKM(ν′) with ν′ ≤ µ. Itfollows that there are uniquely determined integers aµν ≥ 0 with aµµ = 1and

ch MKM(µ) =∑ν≤µ

aµν ch LKM(ν). (3)

This sum usually involves infinitely many non-zero terms. However, forany given ν′ there are only finitely many ν with ν′ ≤ ν and aµν = 0; soe(ν′) occurs with a non-zero coefficient only for finitely many summandson the right hand side of (3).

One can now invert the matrix of all aµν ; one gets integers bµν with

ch LKM(µ) =∑ν≤µ

bµν ch MKM(ν) (4)

and bµµ = 1.2.3. Set

XKM := µ ∈ h∗KM | µ(hi) ∈ Z for all i

and

XKM+ := µ ∈ XKM | µ(hi) ≥ 0 for all i .

Choose an arbitrary ρKM ∈ XKM+ with ρKM(hi) = 1 for all i. As in 1.7(4)

we introduce the dot notation w•µ = w(µ+ρKM)−ρKM for all w ∈ WKM

and µ ∈ h∗KM ; it does not depend on the choice of ρKM .

In his 1974 paper [Kc1] Kac generalised Weyl’s character formula tosymmetrisable Kac–Moody algebras. He showed first for all λ ∈ XKM

+that

ch LKM(λ) =∑

w∈WK M

det(w) chMKM(w•λ). (1)

The simple module ch LKM(0) is the one dimensional trivial module C.Now a comparison with 2.2(2) yields the denominator formula∏

α∈R+K M

(1− e(−α))mα =∑

w∈WK M

det(w) e(w•0). (2)

Using this we can rewrite (1) as

ch LKM(λ) =

∑w∈WK M

det(w) e(w•λ)∑w∈WK M

det(w) e(w•0)(3)

in perfect analogy to 1.2(2).


The proof in [Kc1] follows to a large extent the ideas in [BGG1], cf. 1.8.However, at one point something new was needed: In order to show thatch LKM(λ) is a linear combination of all ch LKM(w•λ) with w ∈ WKM ,Kac could not use the centre of the enveloping algebra. Instead heintroduced Casimir operators that act on the modules, but do not comefrom the enveloping algebra.

Actually (3) holds also when gKM is not symmetrisable. This wasproved more than ten years later, independently by Kumar in [Ku2] andby Mathieu in [M].

Let me add that Littelmann’s path model (see 1.9) works equally wellfor the modules ch LKM(λ) with λ ∈ XKM

+ over symmetrisable Kac–Moody algebras.

2.4. Assume from now on that gKM is symmetrisable. In this caseKac and Kazhdan determined in [KK] all composition factors of allVerma modules MKM(µ), i.e., all ν with aµν = 0 in 2.2(3). In con-trast to the classical case they could not conclude that aµν = 0 impliesν ∈ WKM •µ. However, if we exclude certain weights, then the situationgets better.

For each positive imaginary root α set

Hα := λ ∈ h∗KM | 2(λ + ρKM , α) = (α, α) . (1)

These Hα are called the critical hyperplanes in h∗KM . The complement

h∗KM \

⋃α Hα of the union of the critical hyperplanes is stable under the

dot action of WKM . For each µ in this complement the equation 2.2(4)has the form

ch LKM(µ) =∑

ν∈WK M •µ

bµν ch MKM(ν). (2)

This was shown in [Ku1] extending partial results in [DGK].A weight λ ∈ XKM

+ does not belong to any critical hyperplane becausewe have (λ + ρKM , α) ≥ (ρKM , α) > 0 for any positive root α, but(α, α) ≤ 0 for any imaginary root α. Therefore (2) can be applied toany µ = w•λ with w ∈ WKM . In this case [DGK] contains a conjecturefor the bµν (or rather for the inverse matrix of the bµν ). This conjecturewas then proved in the two papers [Ks] by Kashiwara and [KT1] byKashiwara and Tanisaki; it was independently proved by Casian in [Cs].The result is that

ch LKM(w•λ) =∑

x∈WK M

det(wx)QKMw,x (1) ch MKM(x•λ) (3)

for all λ ∈ XKM+ and w ∈WKM .


Here the QKMw,x are the inverse Kazhdan–Lusztig polynomials for WKM .

The construction of the Kazhdan–Lusztig polynomials works for all Cox-eter groups, hence also for WKM ; we denote here the Kazhdan–Lusztigpolynomials by PKM

w,x . These polynomials are usually only introducedwhen w x where is the Bruhat-Chevalley order on WKM ; we sethere PKM

w,x = 0 in all other cases.The inverse Kazhdan–Lusztig polynomials QKM

w,x are zero unless w x

and are defined by the equations∑y∈WK M

det(xy)QKMx,y PKM

y ,z = δx,z (4)

for all x, z ∈ WKM . (Note that only y with x y z can contribute anon-zero term to this sum; there are finitely many y with this property.)

If A is the Cartan matrix of the Lie algebra g from 1.1, then PKMx,w is

the polynomial Px,w from 1.10. In this case one can show that QKMw,x =

PKMxw 0 ,ww 0

. Therefore (3) is a generalisation of the result in 1.10, theKazhdan–Lusztig conjecture.

2.5. In the classical case one gets easily the character formula forall L(µ) with µ ∈ X once one knows the character formula for all L(µ)with µ ∈W •X+ . This does not extend to general Kac–Moody algebras.Besides the problem with the critical hyperplanes, also the fact thatXKM is in general not equal to WKMXKM

+ makes trouble. Let us lookat a crucial special case, the affine Kac–Moody algebras.

Consider g as in 1.1 and suppose in addition that g is simple. Denoteagain the simple roots by α1 , α2 , . . . , αr . Let β ∈ R+ ∩X+ be a positiveroot that is also a dominant weight. There are one or two choices for β

depending on whether all roots have the same length or not. If we writeβ =

∑ri=1 niαi with ni ∈ N, then ni > 0 for all i.

We now associate to g and β a generalised Cartan matrix A with indexset I = 0, 1, . . . , r. For i, j ≥ 1 set aij = 2(αi, αj )/(αi, αi); so this partyields the usual Cartan matrix for g. To this add a0i = −2(β, αi)/(β, β)and ai0 = −2(β, αi)/(αi, αi) for all i > 0 as well as a00 = 2. Note thatA is symmetrisable.

Assume now that gKM is the Kac–Moody algebra associated to A. AnyKac–Moody algebra constructed this way is called an affine Kac–Moodyalgebra.

Set δ = αKM0 +

∑ri=1 niα

KMi ∈ h∗

KM . In this subsection I am writingαKM

i for the simple roots in h∗KM in order to avoid confusion. Now an

elementary calculation shows that δ(hi) = 0 for all i, 0 ≤ i ≤ r. Thisimplies si(δ) = δ for all i, hence w(δ) = δ for all w ∈WKM .


As mentioned in 2.1 there is a WKM –invariant non-degenerate bilinearform on h∗

KM . Now δ(hi) = 0 implies (δ, αKMi ) = 0 for all i, hence

(δ, δ) = 0. Any λ ∈ XKM+ satisfies (λ + ρKM , αKM

i ) > 0 for all i, hence(λ + ρKM , δ) > 0. This implies for all w ∈ WKM

(w•λ + ρKM , δ) = (w(λ + ρKM), w(δ)) = (λ + ρKM , δ) > 0.

So we can never reach any µ with (µ + ρKM , δ) < 0.It can be shown that δ belongs to the root system RKM . In fact, all

mδ with m ∈ Z, m = 0 are roots; they are precisely the imaginary rootsof gKM . It follows that there is only one critical hyperplane in this case,equal to Hδ = µ ∈ h∗

KM | (µ + ρKM , δ) = 0 .There is actually one more family of affine Kac–Moody algebras not

covered by the construction above. If g is of type Cr with r ≥ 1, thenone takes above as β also half the largest root in R. Then one has towrite 2β =

∑ri=1 niαi and to set δ = 2αKM

0 +∑r

i=1 niαKMi ; otherwise

no change is needed.2.6. Keep the assumptions on gKM from 2.5. We could have defined

XKM+ in 1.2 also as the set of all λ ∈ XKM with (λ + ρKM) (hi) > 0 for

all i since ρKM(hi) = 1 for all i. We now set

XKM− := λ ∈ XKM | (λ + ρKM) (hi) < 0 for all i . (1)

Any λ ∈ XKM− satisfies (λ + ρKM , δ) < 0, hence (µ + ρKM , δ) < 0 for

all µ ∈ WKM •λ. Therefore ch LK (µ) is certainly not determined bythe results from 2.4. However, one can apply 2.4(1) to any µ = w•λ

with w ∈ WKM . The coefficients bµν in this case were determined byKashiwara and Tanisaki in the paper [KT2] from 1995; they showed that

ch LKM(w•λ) =∑

x∈WK M

det(wx)PKMx,w (1) ch MKM(x•λ). (2)

This confirms a conjecture by Lusztig from [Lu3] for the case where allroots in R have the same length.

One gets from (2) and 2.4(3) character formulae for all LKM(µ) withµ ∈ X and (µ + ρKM , δ) = 0. If (µ + ρKM , δ) > 0, then one can findλ ∈ −ρKM +XKM

+ and w ∈ WKM with µ = w•λ; if w has maximal lengthfor this property, then 2.4(3) extends to this case. If (µ + ρKM , δ) < 0,then one can find λ ∈ ρKM + XKM

− and w ∈ WKM with µ = w•λ; if w

has minimal length for this property, then (2) extends to this case. Thisfollows using translation functors, cf. [Ku3] for the second case.

For µ ∈ h∗KM with µ /∈ X and (µ + ρKM , δ) = 0 one gets similar

results when one replaces WKM with its subgroup of all w ∈ WKM such


that wµ − µ ∈ ZRKM . This was proved by Kashiwara and Tanisaki ina sequence of papers ([KT4], [KT5], and [KT6]). Alternatively one cannow use Fiebig’s paper [Fi] where he generalises Soergel’s approach for g

mentioned in 1.12.One can express the condition (µ + ρKM , δ) = 0 differently. There

exists a unique element c ∈ h0 +∑r

i=1 Chi ⊂ hKM such that α(c) = 0for all α ∈ RKM . Then the one dimensional subspace Cc is the centreof the Lie algebra gKM . The element c acts as scalar multiplication byµ(c) on any MKM(µ) and LKM(µ). Now (µ + ρKM , δ) = 0 is equivalentto µ + ρKM ∈

∑ri=0 Cαi , hence to (µ + ρKM) (c) = 0. So the simple

modules LKM(µ) not covered by the results above are exactly those onwhich c acts as multiplication by −ρKM(c). This value is often calledthe critical level; the number ρKM(c) is a positive integer called the dualCoxeter number of gKM .

2.7. The proofs given by Kashiwara and Tanisaki for the characterformulae 2.4(3) and 2.6(2) have many features in common with the proofof the original Kazhdan–Lusztig conjecture sketched in 1.11. Again flagvarieties and D–modules are involved.

Let us exclude the finite dimensional case. One associates to gKM aKac–Moody group GKM and subgroups B+

KM , B−KM that play the role of

Borel subgroups. These groups can be regarded as group schemes or asinfinite dimensional complex Lie groups.

Set BKM := GKM/B+KM ; this is the (infinite dimensional) flag vari-

ety for GKM . It contains Schubert cells BKM ,w := B+KMwB+

KM/B+KM

with w ∈ WKM . Each BKM ,w is isomorphic to an affine space of dimen-sion l(w) and its closure BKM ,w is the finite union of all BKM ,x withx w.

One can also consider BwKM := B−

KMwB+KM/B+

KM . Each BwKM is iso-

morphic to an infinite dimensional affine space. It has codimension l(w);its closure is the infinite union of all Bx

KM with w x.Now the proof of the character formula 2.6(2) — where we can restrict

the sum to x with x w — involves the geometry of BKM ,w whereasthe proof of 2.4(3) — where we can restrict the sum to x with w x —involves the geometry of Bw

KM .The link between gKM –modules and the geometry of BKM is provided

by suitable categories of D–modules on BKM . The construction of thesecategories together with the appropriate functors to gKM –modules andto Dn

cs(BKM) is quite lengthy and requires some delicate limit construc-tions. You can find more details in the survey [KT3] by Kashiwara andTanisaki themselves.


2.8. Let me conclude this section with a character formula thatfollows from 2.6(2) and that is used when comparing character formulaefor Kac-Moody algebras and quantum groups in Section 3.

Suppose that gKM is an affine Lie algebra constructed as in 2.5 start-ing from a finite dimensional simple complex Lie algebra g. Then onecan identify the root system R of g with the subset RKM ∩

∑ri=1 Cαi .

Furthermore we can identify g with the subalgebra of gKM generated byall gKM ,α with α ∈ R. Under this embedding of g into gKM each gα withα ∈ R is identified with gKM ,α and h is identified with

∑ri=1 Chi ⊂ hKM ;

in fact each hi ∈ h is identified with hi ∈ hKM .We can find an element d ∈ hKM such that α0(d) = 1 and αi(d) = 0

for all i > 0. We have then hKM = h⊕Cc⊕Cd and we can characterise R

as the set of all α ∈ RKM with α(d) = 0. For all i ∈ Z set

g[i]KM := x ∈ gKM | [d, x] = i x . (1)

We get then g[0]KM = g⊕Cc⊕Cd. Any g

[i]KM with i = 0 is the direct sum

of all root spaces gKM ,α with α(d) = i. The explicit description of theroot system RKM implies that all g

[i]KM are finite dimensional. We have

[g[i]KM , g

[j ]KM ] ⊂ g

[i+j ]KM for all i and j.

Set

q+ :=⊕i>0

g[i]KM , q− :=

⊕i<0

g[i]KM , p :=

⊕i≥0

g[i]KM = g

[0]KM ⊕q+ . (2)

These are subalgebras of gKM with gKM = q− ⊕ p as a vector space.Set

Y := µ ∈ h∗KM | µ(hi) ∈ N for all i ≥ 1 . (3)

Any µ ∈ Y defines a finite dimensional simple g–module V (µ|h) with

highest weight µ|h. We extend V (µ|h) to a g[0]KM module VKM(µ) by

letting c act as multiplication by µ(c) and d act as multiplication byµ(d). (Recall that c and d commute with g.) We extend VKM(µ) furtherto a p–module by letting q+ act as 0; this works since q+ is an ideal in p.

Now consider the induced gKM –module

M ′KM(µ) := U(gKM)⊗U (p) VKM(µ) (4)

that we call a generalised Verma module. It is clearly a homomorphic im-age of the Verma module MKM(µ). The subspace 1⊗VKM(µ) of M ′

KM(µ)is a finite dimensional g–submodule of M ′

KM(µ) and generates M ′KM(µ)

as a gKM –module. It follows that M ′KM(µ) considered as a g–module is

locally finite, i.e., a sum of finite dimensional g–submodules. (If V is a


finite dimensional g–submodule in a gKM –module M , then each g[i]KM V

is a finite dimensional g–submodule. Therefore the sum of all finitedimensional g–submodules in M is a gKM –submodule of M .)

Now also all subquotients of the gKM –module M ′KM(µ) are locally

finite for g. This applies in particular to all composition factors. IfLKM(µ′) is a composition factor of M ′

KM(µ), then we get µ′ ∈ Y . Other-wise there exists i ≥ 1 with µ(hi) /∈ N; then the g–module generatedby a highest weight vector v in LKM(µ′) is infinite dimensional since itcontains the infinitely many linearly independent vectors ym v, m ≥ 0,with y ∈ g−αi

, y = 0.One gets now as in 2.2 that there exist integers cµν with cµµ = 1 such

that

ch LKM(µ) =∑

ν∈Y ,ν≤µ

cµν ch M ′KM(ν) (5)

for any µ ∈ Y .One can identify the Weyl group W of R with the subgroup of WKM

generated by all si with i ≥ 1. One has then (wλ)|h = w(λ|h) and(w•λ)|h = w•(λ|h) for all w ∈ W and λ ∈ h∗

KM . For any µ ∈ Y we canwrite 1.7(3) in the form ch V (µ|h) = ch U(n−)

∑w∈W det(w)e(w•µ|h).

This implies ch VKM(µ) = ch U(n−)∑

w∈W det(w)e(w•µ). (Use that(wµ) (c) = µ(c) and (wµ) (d) = µ(d) for w ∈W .) One gets now

ch M ′KM(µ) = ch U(q−) ch VKM(µ) =

∑w∈W

det(w) ch MKM(w•µ) (6)

for all µ ∈ Y .Suppose now that µ ∈ Y with µ(h0) ∈ Z and (µ+ ρKM , δ) < 0. There

exists z ∈ WKM such that λ = z−1•µ satisfies (λ + ρKM) (hi) ≤ 0 forall i, 0 ≤ i ≤ r. Choose z of minimal length with this property. Thenone gets from the generalisation of 2.6(2) that

ch LKM(µ) = ch LKM(z•λ) =∑

x∈WK M

det(xz)PKMx,z (1) ch MKM(x•λ).

Comparing with (5) and (6) we get for any ν ∈ Y that cµν is equal tothe sum of all det(xz)PKM

x,z (1) with x ∈ WKM and x•λ = ν. We get thus

ch LKM(z•λ) =∑

x∈WK M , x•λ∈Y

det(xz)PKMx,z (1) ch M ′

KM(x•λ). (7)


3 Quantum groups

The quantum groups that we consider here should more properly becalled quantised enveloping algebras. One can associate such an alge-bra to any symmetrisable Kac–Moody algebra. However, we restrictourselves here to quantum groups corresponding to finite dimensionalsemi-simple complex Lie algebras. For more information on quantumgroups we refer to [CP], [Ja2], [Jo2], and [Lu4].

3.1. Let g be a finite dimensional semi-simple complex Lie algebra asin 1.1. We keep the notations like h, R, R+ , X, X+ , W from 1.1. In par-ticular α1 , α2 , . . . , αr will be a basis for the root system R. We supposethat the W–invariant form on h∗ is normalised such that (α, α) = 2 forall short roots in any irreducible component of R. This normalisationimplies that (λ, α) ∈ Z for all α ∈ R and λ ∈ X. Set di = (αi, αi)/2 forall i; we have then di ∈ 1, 2, 3.

Fix a complex number q that is transcendental over the rational num-bers. Set qi = qdi for all i. Now the quantised enveloping algebra Uq (g)is the associative algebra over C with 4r generators Ei, Fi,Ki,K

−1i

(1 ≤ i ≤ r) and certain relations.Among these relations there are the requirements that each K−1

i isindeed the multiplicative inverse to Ki and that each Ki commutes witheach Kj . So the subalgebra U 0

q (g) generated by all Ki and K−1i is

commutative. One can show that all Km 11 Km 2

2 . . . Kmrr with

(m1 ,m2 , . . . ,mr ) ∈ Zr

form a basis for U 0q (g). Any λ ∈ X defines an algebra homomorphism

from U 0q (g) to C that we again denote by λ:

λ : U 0q (g) −→ C, Km 1

1 Km 22 . . . Kmr

r → q(λ,m 1 α1 +m 2 α2 +···+mr αr ) . (1)

Now some of the other defining relations for Uq (g) can be formulatedas KEiK

−1 = αi(K)Ei and KFiK−1 = αi(K)−1 Fi = (−αi)(K)Fi for

all i and all K ∈ U 0q (g).

The remaining relations are the quantum Serre relations (see the ref-erences) and the condition that EiFj −FjEi = δij (Ki−K−1

i )/(qi−q−1i )

for all i and j.Let U+

q (g) be the subalgebra of Uq (g) generated by all Ei and U−q (g)

the subalgebra generated by all Fi . One gets then a triangular decom-position of Uq (g): The multiplication map induces an isomorphism

U−q (g)⊗ U 0

q (g)⊗ U+q (g) ∼−→ Uq (g) (2)

of vector spaces.


3.2. If V is a U 0q (g)–module, then one defines weight spaces

Vλ := v ∈ V | u v = λ(u) v for all u ∈ U 0q (g) (1)

for all λ ∈ X. The sum of these Vλ is direct. If all Vλ are finite dimen-sional and if V is equal to their sum, then one defines a formal characterch V as in 1.2.

For example, U+q (g) and U−

q (g) are U 0q (g)–modules such that each Ki

acts by conjugation: we have Ki · x = Ki xK−1i for any x in U+

q (g) orU−

q (g). For this action one gets

ch U+q (g) =

∑µ∈X

P (µ) e(µ) and ch U−q (g) =

∑µ∈X

P (−µ) e(µ) (2)

where P is Kostant’s partition function from 1.6. In other words, wehave ch U+

q (g) = ch U(n+) and ch U−q (g) = ch U(n−) where we regard

U(n+) and U(n−) as h–modules via the adjoint action.One can show that there exists for each dominant weight λ ∈ X+

a finite dimensional simple Uq (g)–module Vq (λ) generated by a highestweight vector vλ of weight λ. This means that Ei vλ = 0 for all i andthat K vλ = λ(K) vλ for all K ∈ U 0

q (g). Then Vq (λ) is the direct sumof all Vq (λ)µ with µ ∈ X and it turns out that the character of Vq (λ) isgiven by Weyl’s character formula: We have

ch Vq (λ) = ch V (λ) (3)

for all λ ∈ X+ .The Vq (λ) do not exhaust the isomorphism classes of finite dimensional

simple Uq (g)–modules. One gets for each τ = (τ1 , τ2 , . . . , τr ) ∈ ±1r aone dimensional Uq (g)–module Cτ where each Ki acts as multiplicationby τi whereas all Ei and Fi act as 0.

An arbitrary finite dimensional simple Uq (g)–module is isomorphic toa tensor product Cτ ⊗ Vq (λ) with λ ∈ X+ and τ as above. For thistensor product to make sense we have to introduce a comultiplication∆: Uq (g) → Uq (g)⊗Uq (g) on Uq (g). It can be chosen such that ∆(Ei) =Ei ⊗ 1 + Ki ⊗Ei and ∆(Fi) = Fi ⊗K−1

i + 1⊗Fi and ∆(Ki) = Ki ⊗Ki

for all i. There are also a counit and an antipode on Uq (g) giving it thestructure of a Hopf algebra.

3.3. The main object of our interest in this section is not Uq (g)itself, but a modification where q is replaced by a root of unity. Thereare two quite different ways of constructing such a quantum group at aroot of unity and the algebras one gets have quite different properties.We shall here look at Lusztig’s version of the construction that involves


divided powers. (For a survey on the other version, first investigated byDe Concini and Kac, see [DCP].)

For any n ∈ Z set [n]i := (qni − q−n

i )/(qi − q−1i ) for 1 ≤ i ≤ r; this is

a non-zero element in Z[qi, q−1i ] except for [0]i = 0. In case n ≥ 0 set

[n]!i := [1]i [2]i . . . [n]i ; this means in particular that [0]!i = 1. One thencalls all E

(n)i := En

i /[n]!i and F(n)i := Fn

i /[n]!i with n ∈ N and 1 ≤ i ≤ r

divided powers.Set A = Z[q, q−1 ]. Let Uq (g)A denote the A–subalgebra of Uq (g)

generated by all E(n)i and F

(n)i as above together with all K±1

i . One canshow that Uq (g)A contains all elements of the form[

Ki ; an

]:=

n∏j=1

Kiqa−j+1i −K−1

i q−(a−j+1)i

qji − q−j

i

(1)

with a ∈ Z and n ∈ N.3.4. For any ζ ∈ C, ζ = 0, set

Uζ (g) := Uq (g)A ⊗A C (1)

where we regard C as an A–algebra with the unique ring homomorphismA → C such that q → ζ.

By abuse of notation we write E(n)i for the element E

(n)i ⊗ 1 ∈ Uζ (g).

We use a similar convention for F(n)i , Ki , and

[Ki ;a

n

]. Set U+

ζ (g) resp.

U−ζ (g) equal to the subalgebra of Uζ (g) generated over C by all E

(n)i

resp. all F(n)i with n ∈ N. Set U 0

ζ (g) equal to the subalgebra generated

by all K±1i and all

[Ki ;a

n

]as above. Then the multiplication map induces

an isomorphism

U−ζ (g)⊗ U 0

ζ (g)⊗ U+ζ (g) ∼−→ Uζ (g) (2)

of vector spaces.The Hopf algebra structure on Uq (g) induces first a Hopf algebra struc-

ture on Uq (g)A and then one on Uζ (g).3.5. The algebra U 0

ζ (g) is commutative. Any λ ∈ X induces analgebra homomorphism U 0

ζ (g) → C, again denoted by λ, such that

λ(Ki) = ζdi λ(hi ) and λ([Ki ; a

n

]) =

[λ(hi) + a

n

]i, ζ

where[

mn

]i, ζ

denotes the image of[

mn

]i∈ A under the homomor-

phism A → C with q → ζ. (One shows first that the homomorphism


from 3.1(1) maps Uq (g)A ∩ U 0q (g) to A. Then one observes that U 0

ζ (g)can be identified with (Uq (g)A ∩ U 0

q (g))⊗A C.)The induced homomorphism U 0

ζ (g) → C determines λ uniquely. Wecan thus identify X with a subset of HomC–alg(U 0

ζ (g),C). We define foreach U 0

ζ (g)–module V weight spaces Vλ in analogy to 3.2(1), similarlyfor ch V .

Given λ ∈ X+ and a highest weight vector vλ in Vq (λ) as in 3.2 set

Vq (λ)A := Uq (g)A vλ and Vζ (λ) := Vq (λ)A ⊗A C. (2)

Then Vζ (λ) is a Uζ (g)–module satisfying ch Vζ (λ) = ch V (λ). Each Vζ (λ)has a unique simple factor module Lζ (λ).

Each τ = (τ1 , τ2 , . . . , τr ) ∈ ±1r yields also for Uζ (g) a one dimen-

sional module Cτ where any Ki acts as τi and any[

Ki ;an

]as τn

i

[an

]i,ζ

.

An arbitrary finite dimensional simple Uζ (g)–module has then the formCτ ⊗ Vζ (λ) with λ ∈ X+ and τ as above.

3.6. If ζ is not a root of unity, then all Vζ (λ) are simple and thecharacter of Lζ (λ) = Vζ (λ) is given by Weyl’s formula.

So suppose now that ζ is a root of unity. There is an integer ≥ 1 suchthat ζ is a primitive –th root of unity. Let us assume that > 1 andthat is prime to all non-zero entries of the Cartan matrix of g. Thismeans that is odd and that it is prime to 3 in case g has a componentof type G2 .

In this case one can show for any λ ∈ X+ that all composition factorsof Vζ (λ) have the form Lζ (µ) with µ ∈ X+ ∩Wa •λ. Here Wa is theaffine Weyl group of R (or rather of the dual root system), i.e., the groupof affine transformations of h∗ generated by W and by all translations byelements in ZR. Furthermore we define w•ν for any ν ∈ h∗ as follows:If w ∈ W , then w•ν = w•ν; if w is the translation by γ ∈ ZR, thenw•ν = ν + γ.

As in previous situations it now follows that there are integers bµν forany µ ∈ X+ with bµµ = 1 such that

ch Lζ (µ) =∑

ν

bµν ch Vζ (ν) =∑

ν

bµν ch V (ν) (1)

where ν runs over all ν ∈ X+ ∩Wa •µ with ν ≤ µ.Note that −ζ is a primitive 2–th root of unity under our assumptions.

Andersen has shown in [A3] that

ch L−ζ (µ) = ch Lζ (µ) (2)

for all µ ∈ X+ .


3.7. Continue to assume that ζ is a primitive –th root of unitywith > 1 odd and prime to 3 in case G2 is involved.

Set

C− := λ ∈ X | − ≤ (λ + ρ) (hα ) ≤ 0 for all α ∈ R+ . (1)

This is a fundamental domain for the •–action of Wa on X. So anyµ ∈ X can be written in the form µ = x•λ with λ ∈ C− and x ∈ Wa .We shall below assume that we do so with x of minimal length.

Here the term minimal length refers to a certain generating systemthat turns Wa into a Coxeter group. Our choice here consists of thoseelements that act as reflections with respect to the “walls” of C−. Thismeans explicitly that we take all simple reflections si , 1 ≤ i ≤ r, from W

together with the affine reflections sβ,−1 where β runs over the dominantshort roots of the irreducible components of the root system R. (Bydefinition sβ,−1 is given by sβ,−1(ν) = sβ (ν) − β = ν − (ν(hβ ) + 1)β,hence satisfies sβ,−1•ν = ν − ((ν + ρ) (hβ ) + )β.)

This choice of generating system determines also Kazhdan–Lusztigpolynomials Px,w for all x,w ∈ Wa . In case x,w ∈ W these polynomialsconstructed working with Wa coincide with those constructed workingwith W . This is why we do not introduce a new notation.

Now one can show (see 3.8) under mild restriction for : Let µ ∈ X+

and λ ∈ C− and w ∈ Wa such that µ = w•λ. If w has minimal lengthfor µ = w•λ, then

ch Lζ (w•λ) =∑

x∈Wa , x• λ∈X+

det(wx)Px,w (1) ch V (x•λ). (2)

Here I write det(y) for the determinant of the linear part of any y ∈Wa :If y = y1y2 with y1 ∈W and y2 a translation, then det(y) = det(y1).

This character formula was conjectured by Lusztig in [Lu2] in case allirreducible components of R have only one root length.

3.8. The character formula 3.7(2) follows from the character for-mula 2.8(7) [proved by Kashiwara and Tanisaki] thanks to work byKazhdan and Lusztig in [KL3–6] and [Lu5]. Here I can only say verylittle about this work.

One reduces easily to the case where g is simple. Assume for the mo-ment in addition that all roots in R have the same length, i.e., that R isof type A, D, or E. Let β denote the unique dominant root in R. Con-sider the affine Kac–Moody algebra gKM constructed by the procedurefrom 2.5 working with this choice of β.

We identify g as in 2.8 with a subalgebra of gKM and h with a subspace


of hKM . Furthermore W is identified with a subgroup of WKM . Nowone shows in the theory of affine Kac–Moody algebras that WKM isisomorphic to the affine Weyl group Wa . An isomorphism takes any si

with i ≥ 1 considered as an element in WKM to si considered as anelement in Wa , and it takes s0 to sβ,−1 .

We use this isomorphism to identify Wa and WKM . One checks thatthis is compatible with the definition of det(y) for y ∈ Wa follow-ing 3.7(2). Furthermore, since the identification is compatible with thegenerating sets used to define the Kazhdan–Lusztig polynomials, we getalso

Px,w = PKMx,w for all x,w ∈ Wa . (1)

Set h∗KM ,− := µ ∈ h∗

KM | (µ + ρKM)(c) = − with c as in 2.6. Thenh∗

KM ,− is stable under the dot action of WKM since αi(c) = 0 for all i,0 ≤ i ≤ r. Now an elementary calculation shows that

(w•µ)|h = w•(µ|h) for all µ ∈ h∗KM ,− and w ∈WKM . (2)

(It suffices to take w = si with 0 ≤ i ≤ r.)Consider now λ ∈ C− ⊂ h∗. We can find an element λ ∈ h∗

KM suchthat (λ + ρKM) (c) = − and λ|h = λ. (Recall that hKM = h⊕Cc⊕Cd;so we choose (λ+ρKM) (d) arbitrarily.) We have under our assumptions

(λ+ρKM) (h0) = (λ+ρKM) (c)− (λ+ρKM) (hβ ) = −− (λ+ρ) (hβ ) ≤ 0,

hence (λ + ρKM) (hi) ≤ 0 for all i, 0 ≤ i ≤ r. Therefore 2.8(7) holds(with λ replaced by λ) for all z ∈ WKM with z•λ ∈ Y .

The definition of Y in 2.8(3) shows that z•λ ∈ Y if and only if(z•λ)|h ∈ X+ . Set C equal to the category of all gKM –modules M of finitelength such that all composition factors of M have the form LKM(w•λ)with w ∈ WKM and w•λ ∈ Y . Suppose that we have an exact func-tor F from C to the category of finite dimensional Uζ (g)–modules takingM ′

KM(µ) to Vζ (µ|h) and LKM(µ) to Lζ (µ|h) for each µ ∈ Y ∩WKM •λ.Then F induces a homomorphism of Grothendieck groups, and 2.8(7)implies 3.7(2) using (1) and (2) since we can regard these characterformulae as identities in the Grothendieck groups.

Kazhdan and Lusztig basically construct such a functor in [KL3–6].However there are some differences. First of all, they do not workwith gKM , but with a subalgebra g′KM of codimension 1. This subalgebrais the direct sum of all gKM ,α with α ∈ RKM and of h′

KM := h⊕Cc. Sothey drop the element d ∈ hKM . One now has first to check that thegKM –modules LKM(µ) still are irreducible when restricted to g′KM and


that 2.8(7) is also a character formula for simple g′KM –modules. For thissee Prop. 8.1 in [So4]; the result goes back to Polo.

Then Thm. 38.1 in [KL6] says that there is an equivalence of categoriesbetween suitable categories of g′KM –modules and of U−ζ (g)–modules. (Intype E the proof in [KL6] requires that is not too small; it suffices toassume > 31, see [KL6], Cor. 2 to Lemma 31.5.) According to 9.1in [Lu5] this equivalence takes M ′

KM(µ) to V−ζ (µ|h) for each µ ∈ Y with(µ + ρKM) (c) = −. It then has to take the simple head LKM(µ) toL−ζ (µ|h). One gets now a character formula for simple U−ζ (g)–modules.To conclude the argument one applies 3.6(2).

In case R has two root lengths, this procedure has to be modified, see[Lu5], 9.2. In the end one has to use a generalisation of 2.6(2) where theweights no longer belong to XKM .

4 Prime characteristic

Finally we turn to the analogues in characteristic p > 0 of the complexsemi-simple groups and their Lie algebras. Their representations havebeen investigated for a longer period that those of Kac–Moody algebrasor quantum groups. However, our present knowledge in prime charac-teristic is weaker than that in the other areas, and some of the strongestresults in this area have been proved by comparing prime characteristicand quantum groups at a root of unity. — For more background onalgebraic groups let me refer you as in 1.3 to [Bo1], [Hu2], and [Sp3];for more information on representations of algebraic groups in primecharacteristic one may consult [Ja3].

4.1. Let k be an algebraically closed field of characteristic p > 0. Weconsider a connected and simply connected algebraic group Gk over k;denote its Lie algebra by gk and choose a maximal torus Tk in Gk .We assume that Gk has the same type as our complex Lie algebra g.This means that the root system of Gk with respect to Tk (the non-zeroweights of Tk on gk for the adjoint representation) can be identified withthe root system R of g. The assumption that Gk is simply connected im-plies that we can identify the group of characters of Tk with the group X

of integral weights from 1.1.One can regard Gk as a universal Chevalley group as described (e.g.)

in [Hu1], 27.4. In particular one can identify gk with gZ ⊗Z k where gZ

is the Z–span of a Chevalley basis for g as in [Hu1], 25.2.4.2. A Gk –module is a vector space V over k with a homomorphism


Gk → GL(V ) of algebraic groups. Then V is the direct sum of itsweight spaces Vλ = v ∈ V | t v = λ(t) v for all t ∈ Tk . If V is finitedimensional, then we can define a formal character ch V ∈ Z[X] asbefore. This character is W–invariant.

One can now show: Any simple Gk –module is finite dimensional. Forany dominant µ ∈ X+ there exists a simple Gk –module Lk (µ) such thatdim Lk (µ)µ = 1 and such that Lk (µ)ν = 0 implies ν ≤ µ. These twoproperties determine Lk (µ) uniquely up to isomorphism. We call Lk (µ)the simple Gk –module with highest weight µ. Every simple Gk –moduleis isomorphic to Lk (µ) for exactly one µ ∈ X+ .

One can construct these simple modules using reduction modulo p: In-side the simple g–module V (µ) with highest weight µ one chooses a mini-mal admissible lattice VZ(µ) as in [Hu1], 27.3. Then Vk (µ) := VZ(µ)⊗Z k

can be regarded as a Gk –module via the identification of Gk with aChevalley group. Now Lk (µ) is the unique simple quotient of Vk (µ).The character of Vk (µ) is given by Weyl’s character formula since theconstruction yields

ch Vk (µ) = chV (µ) for all µ ∈ X+ . (1)

One usually calls Vk (µ) the Weyl module with highest weight µ.4.3. For any µ ∈ X+ the composition factors of the Weyl modu-

le Vk (µ) are Lk (µ) with multiplicity 1 and certain Lk (ν) with ν ∈ X+

and ν < µ. This implies that ch Lk (µ) is a Z–linear combination of thech Vk (ν) = ch V (ν) with ν ≤ µ.

One can show more precisely that all composition factors of Vk (µ)have the form Lk (ν) with ν ∈ X+ ∩ Wa •pµ. Here Wa is the affineWeyl group of R as in 3.6 and the •p–action is defined as the •–actionfrom 3.6 taking = p there, i.e., we have w•pν = w•ν for w ∈ W whereasw•pν = ν + pγ in case w is the translation by γ ∈ ZR. So we get asin 3.6: There are integers bµν for any µ ∈ X+ with bµµ = 1 such that

ch Lk (µ) =∑

ν

bµν ch V (ν) (1)

where ν runs over all ν ∈ X+ ∩Wa •pµ with ν ≤ µ.In 1979 Lusztig stated in [Lu1] a conjecture for the coefficients bµν

provided that µ is not “too large” and belongs to a set Mp that we nowdefine. (The need for such a restriction on µ will be explained in 4.4.)Let h denote the maximum of all ρ(hα ) + 1 with α ∈ R+ ; this is theCoxeter number of the root system R. For example, if Gk = SLn (k),


then h = n. Set now

Mp := µ ∈ X+ | (µ + ρ)(hα ) ≤ p(p− h + 2) for all α ∈ R+ . (2)

Note that (µ+ρ)(hα ) > 0 for all α ∈ R+ ; so we haveMp = ∅ if p ≤ h−2.As in 3.7 the set

C− := λ ∈ X | −p ≤ (λ + ρ) (hα ) ≤ 0 for all α ∈ R+

is a fundamental domain for the •p–action of Wa on X. Lusztig’s con-jecture says now: Write µ ∈ X+ in the form µ = w•pλ with λ ∈ C− andw ∈ Wa such that w has minimal length for µ = w•pλ. If w•pλ ∈ Mp ,then

ch Lk (w•pλ) =∑

x∈Wa , x•p λ∈X+

det(wx)Px,w (1) ch V (x•pλ). (3)

Here as in 3.7 “minimal length” refers to the generating system of Wa

given by the reflections with respect to the walls of C−; similarly theKazhdan–Lusztig polynomials are taken with respect to this system.

Now a comparison with 3.7(2) shows that we can reformulate Lusztig’sconjecture as follows: If ζ is a primitive p–th root of unity, then

ch Lk (µ) = ch Lζ (µ) for all µ ∈Mp . (4)

Of course, when Lusztig made his conjecture, quantum groups were stillunknown. At that time it was known that (3) holds for groups of rankat most 2 as well as for Gk = SL4(k).

In [AJS] it was proved that (4) holds if p is large enough. Moreprecisely, given a root system R there exists an (unknown) bound m(R)such that (4) holds whenever p > m(R). We got this type of result byshowing that there are systems of linear equations with coefficients in Zso that (4) holds if and only if the dimensions of the solution spaces donot change when one reduces the coefficients modulo p.

4.4. Recall that α1 , α2 , . . . , αr is our base for R. Set

Xp := µ ∈ X+ | µ(hαi) < p for all i, 1 ≤ i ≤ r . (1)

Each µ ∈ X+ has a unique decomposition µ = pµ1+µ0 with µ0 ∈ Xp andµ1 ∈ X+ . In this situation Steinberg’s tensor product theorem impliesthat

Lk (µ) Lk (pµ1)⊗ Lk (µ0). (2)

For any χ =∑

ν∈X aν e(ν) ∈ Z[X] write χ(p) =∑

ν∈X aν e(pν) ∈ Z[X].


(Here we have aν ∈ Z for all ν.) Another consequence of Steinberg’stensor product theorem is

ch Lk (pµ) =(ch Lk (µ)

)(p)for all µ ∈ X+ . (3)

So we get for µ = pµ1 + µ0 as above

ch Lk (pµ1 + µ0) =(ch Lk (µ1)

)(p)ch Lk (µ0). (4)

This implies: If we know all ch Lk (µ) with µ ∈ Xp , then we cancompute inductively all ch Lk (µ) with µ ∈ X+ .

An elementary calculation shows: If p ≥ 2h− 3, then Xp is containedin the set Mp from 4.3(2). So Lusztig’s conjecture 4.3(3) leads to (con-jectural) character formulae for all ch Lk (µ) with µ ∈ X+ as long asp ≥ 2h− 3.

Lusztig has proved an analogue to Steinberg’s tensor product theoremfor quantum groups. If ζ is a primitive p–th root of unity and if µ =pµ1 + µ0 as above, then Lusztig’s result implies

ch Lζ (pµ1 + µ0) =(ch V (µ1)

)(p)ch Lζ (µ0). (5)

A comparison of (5) and (4) indicates the need for a restriction on w•pλ

in 4.3(3): If we assume to start with that 4.3(4) holds for all µ0 ∈ Xp ,i.e., that ch Lk (µ0) = ch Lζ (µ0) for all these µ0 , then it will hold for µ =pµ1 +µ0 as above if and only if ch Lk (µ1) = chV (µ1). Now the definitionof Mp was done in such a way that the equality ch Lk (µ1) = ch V (µ1)holds for all µ ∈Mp . (Of courseMp was defined before quantum groupswere introduced; the original justification for this bound takes longer toexplain.)

One could now hope that 4.3(4) might hold for all µ ∈ Xp , and forp > h this seems to be a realistic conjecture. However, for smaller primesthings go wrong: Andersen and I found an example for G = SLp+3 where4.3(4) fails for some µ ∈ Xp , see [A2], 7.9. Note that in this exampleh = p+3 > p. At this point there is no conjecture that predicts characterformulae for all Lk (µ) with µ ∈ Xp when p is small.

4.5. The set Xp has another important property: The Lk (µ) withµ ∈ Xp are also simple as modules for the Lie algebra gk of Gk . Infact, a theorem due to Curtis says that the Lk (µ) with µ ∈ Xp are asystem of representatives for the isomorphism classes of simple restrictedgk –modules.

In order to explain the last statement we have to recall that the Lie


algebra of an algebraic group in characteristic p has an extra structure:It comes with a p–th power map x → x[p ] that makes the Lie algebrainto a Lie p–algebra. If one regards x ∈ gk as a derivation (with certaininvariance properties) on the algebra k[Gk ] of regular functions on Gk ,then the p–th power of x is again a derivation (with the same invarianceproperties), hence an element of gk that we denote by x[p ].

We call a representation ϕ : gk → Endk (V ) and the correspondinggk –module V restricted if ϕ satisfies ϕ(x[p ]) = ϕ(x)p for all x ∈ gk . If M

is a Gk –module, then the induced gk –module is automatically restricted.

Regarding all Lk (µ) with µ ∈ Xp as simple gk –modules is an essentialingredient in the proof of 4.3(4) for large p in [AJS]. Lusztig’s conjectureis equivalent to a conjecture on the characters of the projective coversof these Lk (µ) in the category of all restricted gk –modules, and theseprojective covers are the main objects in [AJS]. For a more detailedsurvey of that proof see [So2].

4.6. The approach to Lusztig’s conjecture 4.3(3) described above,via quantum groups and Kac–Moody algebras to the geometry of flagvarieties, goes back to an idea of Lusztig formulated shortly after thediscovery of quantum groups. Of course, this is a rather indirect ap-proach and one may ask whether there is not a more direct way. NowBezrukavnikov has recently announced that one can avoid the tour viaquantum groups, see [Be], Remark 3.13.

Let J0 denote the ideal in U(gk ) generated by all xp−x[p ] with x ∈ gk .So a gk –module is restricted if and only if it is annihilated by J0 . SetI0k equal to the annihilator of the trivial one dimensional gk –module k

in the algebra U(gk )Gk of those elements in the enveloping algebra thatare invariant under the adjoint action of Gk . Over C the analogousalgebra U(g)G coincides with the centre of U(g). But in prime charac-teristic U(gk )Gk is a proper subalgebra of the centre of U(gk ); the centreof U(gk ) contains in addition all xp − x[p ] with x ∈ gk .

Let Ck denote the category of all finite dimensional gk –modules thatare annihilated by I0k and by a power of J0 . A special case of Theo-rem 5.3.1 in [BMR] says: If p is greater than the Coxeter number h of R,then the bounded derived category Db(Ck ) is naturally equivalent withthe bounded derived category Db(Coh(B(1)

k )) of coherent sheaves overOB( 1 )

k. Here Bk is the flag variety of Gk and B(1)

k indicates a Frobenius

twist. So B(1)k coincides with Bk as a topological space with a sheaf of

rings; but we change the k–algebra structure on these rings: If U is an


open subset, the OB( 1 )k

(U) = OBk(U), but any a ∈ k acts on OB( 1 )

k(U)

as p√

a does on OBk(U).

The equivalence of derived categories yields isomorphisms of Grothen-dieck groups K(Ck ) K(Coh(B(1)

k )) K(Coh(Bk )) (Note that one canidentify the categories Coh(B(1)

k ) and Coh(Bk ).) The Grothendieck groupK(Ck ) is (for p > h) a free Z–module of rank |W |; a basis are the classesof the simple modules Lk (w•0 + pρw ) with w ∈ W where ρw ∈ X ischosen such that w•0 + pρw ∈ Xp .

According to [Be], Cor. 3.12 the images of these simple modules inK(Coh(Bk )) are independent of p in case p" 0. For this to make sense,one identifies the complexification K(Coh(Bk )) ⊗Z C with the Borel–Moore homology HBM

• (B) of the flag variety B of G over C. Moreprecisely the claim in [Be] is that the classes of the simple modules aremapped to a “canonical” basis for HBM

• (B) constructed by Lusztig.To get from this result to Lusztig’s conjecture 4.3(3) requires some

work not explained in [Be]. For example, one has to identify the imagesof the projective covers of the simple modules and one has to work withan enriched category where objects are graded by the weight lattice X.

Conclusion

The search for an algebraic proof of Weyl’s character formula — the maintopic of the first section of this survey — has changed our way of lookingat those characters. It is now most natural to think of the character of afinite dimensional simple g–module as a linear combination of “simpler”characters, namely of the characters of Verma modules.

This point of view is then repeated in the generalisations we consider.We always try to express the characters of the simple modules in termsof known characters of “standard” modules. So far one has not foundmore “direct” formulae. Also inductive formulae similar to Freudenthal’sare missing.

In all cases values of Kazhdan–Lusztig polynomials occur as coeffi-cients (up to sign) when we write the character of a simple module asa linear combination of characters of standard modules. These polyno-mials are defined as coefficients expressing one basis of a suitable Heckealgebra in terms of another basis. They can then be computed induc-tively.

The known proofs relate these polynomials rather indirectly to thecharacter formulae. They rely on the fact that the Kazhdan–Lusztig


polynomials describe certain geometric data on flag varieties. On theother hand, the truth of the (Kazhdan–)Lusztig conjectures has usuallystrong applications to the structure of the standard modules. See (e.g.)[Ja3], Section II.C, for the prime characteristic case.

There is an additional case where Kazhdan–Lusztig polynomials (orrather some “periodic” generalisations by Lusztig) play a role. This is forcertain non-restricted representations of gk . In this case we have so faronly experimental evidence from some examples and no general results.But the work in [BMR] and [Be] looks like steps in the right direction.

Bibliography[A1] H. H. Andersen: A new proof of old character formulas, pp. 193–207

in R. Fossum et al. (eds.), Invariant Theory , Proc. Denton, TX, 1986(Contemp. Math. 88), Providence, RI, 1989 (Amer. Math. Soc.)

[A2] H. H. Andersen: Finite-dimensional representations of quantum groups,pp. 1–18 in W. F. Haboush, B. J. Parshall (eds.), Algebraic Groups andtheir Generalizations: Quantum and Infinite-dimensional Methods, Proc.University Park, Penn. 1991 (Proc. Sympos. Pure Math. 56:2), Provi-dence, R. I. 1994 (Amer. Math. Soc.)

[A3] H. H. Andersen: Quantum groups at roots of ±1, Commun. Algebra 24(1996), 3269–3282

[AJS] H. H. Andersen, J. C. Jantzen, W. Soergel: Representations of quantumgroups at a pth root of unity and of semisimple groups in characteristic p:independence of p, Astérisque 220 (1994)

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12The Classification of Affine Buildings

Richard M. WeissDepartment of Mathematics

Tufts [email protected]

1 Introduction

The theory of affine buildings reveals fascinating links between grouptheory, Euclidean geometry and number theory. In particular, reflec-tions, the Weyl chambers of root systems and valuations of fields allplay a central role in their classification.

The study of affine buildings was begun by Bruhat and Tits in [4] andthe classification of affine buildings of rank at least four was completedby Tits in [11]. When combined with the classification of Moufang poly-gons carried out in [12], the Bruhat-Tits classification covers also affinebuildings of rank three under the assumption in this case that the build-ing at infinity is Moufang.

Our goal here is to give a very brief overview of this work. All thedetails can be found in the forthcoming book [14] (as well, of course, asin [4] and [11]).

In this article we regard buildings exclusively as certain edge-coloredgraphs. For different points of view, see [1]. Other excellent sources ofresults about affine buildings are [3], [5] and [8].

2 Buildings

Let Σ be an edge-colored graph and let I denote the set of colors ap-pearing on the edges of Σ. We call |I| the rank of Σ. For each subset J

of I let ΣJ be the graph obtained from Σ by deleting all the edges whosecolor is not in J (but without deleting any vertices). A J-residue of Σfor some subset J of I is a connected component of the graph ΣJ . Thustwo distinct J-residues (for a fixed subset J of I) are always disjoint.

271

272 Richard M. Weiss

A panel is a residue of rank one. In other words, a panel is a maximalconnected monochromatic subgraph.

A chamber system is a connected edge-colored graph Σ such that allpanels are complete graphs having at least two vertices. Thus, in par-ticular, for every vertex x of a chamber system and every color i ∈ I,there exists a vertex y such that x, y is an edge of color i. Accordingto convention, the vertices of a chamber system are called chambers. Achamber system is called thin (resp. thick) if all its panels are of sizeexactly two (resp. at least three).

A chamber system of rank one is just a complete graph. Suppose thatΣ is a chamber system of rank two. Let Γ be the graph whose verticesare the panels of Σ, where two panels are joined by an edge whenevertheir intersection contains a chamber. Two panels of the same color arealways disjoint. Thus Γ is bipartite. Since Σ is connected, so is Γ. Sinceevery panel of Σ contains at least two chambers, every vertex of Γ has atleast two neighbors. Suppose, conversely, that Γ is a connected bipartitegraph such that every vertex has at least two neighbors. Then there isa unique decomposition of the vertex set of Γ into “black” and “white”vertices, so that every edge joins two vertices of different colors. LetΣ be the edge-colored graph with color set I = black, white whosevertices are the edges of Γ, where two edges of Γ are joined by a black(resp. white) edge of Σ whenever their intersection contains a black(resp. white) vertex of Γ. Then Σ is a chamber system of rank two.These two constructions are inverses of each other. Thus a chambersystem of rank two is essentially the same thing as a connected bipartitegraph.

Let Π be a Coxeter graph with vertex set I. This means that Π is anedge-colored graph whose edge “colors” are elements in the set

n ∈ Z | n ≥ 3 ∪ ∞.

The “color” of an edge of a Coxeter graph is usually referred to as its“label.” We denote the label on an edge e by me . Associated with Πis the corresponding Coxeter group W . This is the group generated bythe vertex set I of Π that is defined by the relations i2 = 1 for all i ∈ I,(ij)me = 1 for each edge e such that me = ∞, where i and j are theelements of I joined by e, and (ij)2 = 1 for all pairs of distinct elementsi and j of I that are not adjacent in Π. Let ΣΠ be the edge-coloredgraph with vertex set W and color set I, where two elements x and y

of W are joined by an edge of color i whenever y = xi. Then ΣΠ isa chamber system of rank |I| with panels of size two (i.e. ΣΠ is thin).

The Classification of affine buildings 273

Notice that I is now simultaneously the vertex set of Π, a distinguishedgenerating set of W and the set of colors on the edges of ΣΠ .

If, for example, Π has just two vertices and n is the label on its oneedge, then W is the dihedral group D2n of order 2n and ΣΠ is a circuithaving 2n vertices whose edges are colored alternately black and white.

We can now give the definition of a building. Let Π be a Coxeter dia-gram with vertex set I. A building of type Π is a chamber system ∆ withcolor set I containing a distinguished set of subgraphs called apartmentsall isomorphic (as edge-colored graphs) to the chamber system ΣΠ suchthat the following hold:

(i) For every pair of chambers of ∆, there exists an apartment con-taining them both.

(ii) If A1 and A2 are apartments both containing chambers x and y,then there exists a (color-preserving) isomorphism from A1 to A2

fixing x and y.(iii) If A1 and A2 are apartments both containing a chamber x and

both containing chambers in a panel P , then there exists a (color-preserving) isomorphism from A1 to A2 fixing x and mapping A1∩P to A2 ∩ P .

We observe explicitly that nothing is said in the definition of a build-ing about the automorphism group of the building; in particular, theisomorphisms in conditions (ii) and (iii) are not necessarily induced byautomorphisms of ∆.

A building of type Π is called irreducible if the Coxeter diagram Π isconnected. Buildings which are not irreducible can be decomposed asa kind of direct product and studied one “component” at a time. Fromnow on, we will simply assume that all buildings are irreducible, usuallywithout saying this explicitly.

Notice that ΣΠ is itself a building of type Π. This is the unique thinbuilding of type Π. From now on, we will assume that all buildings arethick, usually without saying this explicitly.

Before going on to the next section, we introduce the notion of a root:A reflection of the Coxeter chamber system ΣΠ is a color-preservingautomorphism of ΣΠ fixing an edge. If r is a reflection of ΣΠ , we denoteby Mr the set of edges fixed by r. A wall is a set of edges of ΣΠ ofthe form Mr for some reflection r. Every edge of ΣΠ lies in a uniquewall. Let M = Mr be a wall and let ΣM

Π denote the graph obtained fromΣΠ by deleting all the edges in M but without deleting any chambers.Then the graph ΣM

Π has exactly two connected components, and they


are interchanged by the reflection r. A root of ΣΠ is the set of chambersin a connected component of ΣM

Π for some wall M .If ∆ is a building of type Π, then each apartment is isomorphic to ΣΠ .

It thus makes sense to talk about roots of an apartment of ∆.

3 Spherical Buildings

A Coxeter diagram Π is called spherical if the corresponding Coxetergroup W (i.e. the chamber set of the chamber system ΣΠ) is finite. Abuilding is called spherical if it is of spherical type. Equivalently (sincethe apartments of a building of type Π are all isomorphic to ΣΠ), abuilding is spherical if its apartments are finite.

Suppose that Π is connected and has only two vertices. Then Π isspherical if and only if the label n on its unique edge is finite. Supposethat, in fact, n is finite, and let ∆ be a building of type Π. Then ∆ is achamber system of rank two. Let Γ be the corresponding bipartite graphas described in §1. Then every vertex of Γ has at least three neighbors(i.e. Γ is “thick”), the diameter of Γ (with respect to the metric on thevertex set of Γ which counts the minimal number of edges needed to gofrom one vertex to another) is n and the length of its minimal circuits(i.e. its girth) is 2n. Moreover, all thick bipartite graphs of diameter n

and girth 2n arise in this way.A thick bipartite graph of diameter n and girth 2n is called a general-

ized polygon (or, more specifically, a generalized n-gon). Thus sphericalbuildings of rank two whose Coxeter diagram has label n and generalizedn-gons are essentially the same thing.

Let Γ be a generalized triangle (i.e. a generalized 3-gon). Then Γis, in particular, bipartite, so there is a unique decomposition of thevertex set of Γ into “white” and “black” vertices so that every edge joinsvertices of different color (as was already observed in §1). We now callthe white vertices “points” and the black vertices “lines” and we declarethat a “point” is incident with a “line” if they are connected by an edgeof Γ. In this fashion we obtain a geometry of points and lines in whichany two “points” lie on a unique “line” and any two “lines” contain aunique “point.”1 Thus Γ is the “incidence graph” of a projective plane.

1 Here is the proof: Let x and y be two points. (We omit the quotation marks.)Every path in Γ beginning and ending at points must pass alternately throughpoints and lines and hence has even length. Thus the distance from x to y is even.Since the diameter of Γ is three, the distance from x to y must be two. In otherwords, there is a line z adjacent to both x and y. If z′ were a second such line,then (x, z, y, z′, x) would be a circuit of length four. Since the girth of Γ is six, it


Conversely, the incidence graph of an arbitrary projective plane is ageneralized triangle. Thus generalized triangles and projective planesare essentially the same thing.

The most typical projective planes arise as the set of one-dimensionalsubspaces (the “points”) and two-dimensional subspaces (the “lines”) of aright vector space over a field K (skew or commutative), where incidenceis given by containment. Choose such a vector space V and let Γ be thecorresponding generalized triangle. If v1 , v2 , v3 is a basis of V over K,then the subgraph of Γ spanned by the three “points” 〈v1〉, 〈v2〉 and 〈v3〉and the three “lines” 〈v1 , v2〉, 〈v2 , v3〉 and 〈v3 , v1〉 is an apartment Σ of Γ(i.e. a circuit of length six), and every apartment of Γ arises in this way.The vector space V is, in some sense, constructed from a combinatorialobject, namely the basis v1 , v2 , v3, and an algebraic object, namelythe field K. Thus also the generalized triangle Γ can be viewed as ageometrical object constructed from a combinatorial object, namely theapartment Σ, and an algebraic object, namely the field, which serves totransform the thin apartment Σ into the thick building ∆.

In [9], Tits classified all spherical buildings of rank at least three.In [12], this classification was extended to the case = 2 under theassumption that the spherical building is Moufang.2

The classification of Moufang spherical buildings of rank at least twoshows that these buildings are all like the projective planes describedabove in that they are uniquely determined by the structure of a fixedapartment Σ (i.e. by the Coxeter diagram Π) and an algebraic structureΛ (a kind of “thickening agent”). The algebraic structure Λ depends onthe family of buildings under consideration. For some families it is a fieldbut for others, Λ is a skew-field, an anisotropic quadratic space, a Jordandivision algebra, a composition algebra or one of an assortment of moreexotic structures. In each case, the algebraic structure Λ parametrizesthe root groups associated with all the roots of a fixed apartment Σ of∆, and ∆ is, in turn, uniquely determined by this “root datum.”

follows that z is unique. The proof that every two lines intersect in a unique pointis obtained by simply interchanging the words “point” and “line.”

2 In fact, all irreducible thick spherical buildings of rank least three as well as allthe residues of rank two of such a building are Moufang. This is a consequenceof the main theorem 4.1.2 in [9]; see 11.6 in [13]. Here is the definition of theMoufang condition: Let a be a root of an apartment Σ (as defined at the end of§2) in a spherical building ∆. The root group Ua is the group consisting of allautomorphisms of ∆ which acts trivially on every panel containing two chambersin a. The Moufang property says that for every root a, the root group Ua actstransitively on the set of all apartments of ∆ containing a (among which Σ is onlyone).


In each case, the algebraic structure Λ is defined over a field whichwe denote by K.3 We say, in fact, that the building ∆ itself is definedover K. In some cases, K equals Λ but in some cases it does not. Forexample, when Λ is an anisotropic quadratic space or a Jordan divisionalgebra, K is the field over which the underlying vector space J of Λ isdefined. In these two cases, some of the root groups are isomorphic tothe additive group of K and the others to the additive group J .

Let Ξ denote the set of roots of our fixed apartment Σ. There is alwaysa canonical subset Ξ0 of Ξ such that for each a ∈ Ξ0 , Ua is isomorphicto the additive group of K.4 The subset Ξ0 of Ξ will be important in§5.

4 Affine Buildings

Suppose that Π is the Coxeter diagram with two vertices whose uniqueedge has the label ∞. Then the bipartite graphs corresponding to build-ings of type Π are trees such that every vertex has at least two neighbors,and the apartments of these buildings are the subtrees A such that eachvertex of A has exactly two neighbors in A. Thus we can think of theapartments as one-dimensional affine spaces partitioned by the integers.This is the simplest example of an affine building.

An affine Coxeter diagram is a diagram Π each connected componentof which is one of the irreducible diagrams in Figure 12.1 and an affinebuilding is a building whose Coxeter diagram is affine.

Each diagram in Figure 12.1 has a name of the form X , where X isthe name of one of the spherical diagrams in Figure 12.2. Here is thenumber of vertices of X and also the number of vertices of X minus one.In each case the diagram X can be obtained from X by deleting a singlevertex x and all the edges containing x. (The vertices of the diagram X

with this property are called special. Thus, for example, every vertex ofA for ≥ 2 is special, the special vertices of B are the two at the leftand the special vertices of C are the vertices at the two extremes.)

We call the spherical diagram X the derived diagram of X . For

3 We must, in fact, allow K to be a skew-field or an octonion division ring in somecases. If ∆ is the spherical building associated with the F -rational points of anabsolutely simple algebraic group G of F -rank at least two, then ∆ is Moufangand F is either the center of K or the intersection of the center of K with thefixed point set of a certain involution of K .

4 This statement must be slightly modified for the Moufang quadrangles of indiffer-ent and exceptional type, but we omit the details in this survey. If Π is simplylaced, then Ξ0 = Ξ.


• •.............................................................∞

A1

................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................••••••

• • • •A ( ≥ 2) ...........•

•• • • • •...................................................................................................................................................................................................... ...............................................................................

.....................................................

................................

.....................B ( ≥ 3) ........

• • • • • •................................................................................................................................................................................................................................................................................................................................ .................................................................................C ( ≥ 2) ........

•

•• • • •

•

•

.....................................................

................................

................................................................................................. ............................................................................................

................................

.....

.....................................................D ( ≥ 4) . . . . . . .

• • ••

• •

•

.............................................................................................................................................................................................................................................................................................................................

E6

• • • ••

• • •...............................................................................................................................................................................................................................................................................................................................................................................................................

E7

• • • ••

• • • •...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................

E8

• • •••......................................................................................................................... .......................................................................................................................................................................................F4

• • •.........................................................................................................................6

G2

Fig. 12.1. The Irreducible Affine Coxeter Diagrams

each ≥ 3, the Coxeter diagrams B and C are distinct but have thesame derived diagram. Every other diagram in Figure 12.1 is uniquelydetermined by its derived diagram.

We now fix a connected affine diagram Π = X and let ΣΠ be thecorresponding chamber system. Then ΣΠ has a natural representationin terms of “alcoves” in a Euclidean space V of dimension which wenow describe.

Let Φ be the roots system called X , let V denote the ambient Eu-clidean space, let

B := α1 , α2 , . . . , α


• • • • • •........................................................................................................................................ ...........................................................................................................................................A ( ≥ 1) ........

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Fig. 12.2. Some Spherical Coxeter Diagrams

be a basis of Φ, let

sα (v) = v − 2v · αα · αα

for all α ∈ Φ and all v ∈ V , let

S = sαi| i ∈ [1, ]

and let W = 〈S〉. The group W is the Weyl group of Φ. Up to iso-morphism (W,S) is the Coxeter system corresponding to the deriveddiagram X of X .

For each pair (α, k) ∈ Φ× Z, let

sα,k (v) = sα (v) + 2kα/(α · α)

for all v ∈ V . The elements sα,k are called affine reflections of V . letα be the highest root of Φ with respect to the basis B as defined inProposition 25 in Chapter VI, Section 1.8, of [2], let

S = sαi ,0 | i ∈ [1, ] ∪ sα,1

and let W be the group generated by S. Then (W , S) is isomorphic tothe Coxeter system corresponding to the diagram X .


For each pair (α, k) ∈ Φ×Z, we denote by Hα,k the affine hyperplane

v ∈ V | α · v = k.

Let

X =⋃

(α,k)∈Φ×Z

Hα,k .

An alcove of Φ is a connected component of V \X. One alcove, forexample, is

v ∈ V | v · αi > 0 for all i ∈ [1, ] and v · α < 1.

The Weyl group W acts sharply transitively on the set of alcoves. Eachalcove is bounded by exactly + 1 of the hyperplanes in X. Let Γ bethe graph whose vertices are the alcoves, where two distinct alcoves areadjacent in Γ whenever there is a hyperplane in X bounding both ofthem. Then the edges of Γ can be colored (with color set the vertex setof the Coxeter diagram X) to make Γ isomorphic to ΣΠ .

Here, for example, is how the alcoves look if Π = C2 :

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The affine hyperplanes in the set X are the straight lines in this pictureand the alcoves in this picture are the open triangles. Two alcoves areadjacent when they are separated by a segment in one of the affinehyperplanes, and if two alcoves are separated by a segment, then thecolor of the corresponding edge of ΣΠ is the “color” of the segment,either “dotted,” “dashed” or “solid.”

A point v in the ambient space V of Φ is called special if for eachα ∈ Φ, there exists k ∈ Z such that v ∈ Hα,k . A Weyl chamber of theroot system Φ is a set of the form⋂

1≤i≤

v ∈ V | v · βi > 0

for some basis β1 , β2 , . . . , β of Φ. A sector of ΣΠ is the set of alcovescontained in a fixed translate of a Weyl chamber whose apex is a specialpoint of V . If Π = C2 , for example, then the special points are those at


the intersection of four hyperplanes of the form Hα,k , and the sectors arethe set of alcoves contained in the wedge bounded by two hyperplanesof the form Hα,k making an angle of 45 degrees.

Two sectors of ΣΠ are called parallel of their intersection is also asector. We denote the parallel class of a sector S by S∞ and we denotethe set of all parallel classes of sectors in ΣΠ by Σ∞

Π . Then Σ∞Π has,

canonically, the structure of an apartment in a building of type X . IfΠ = C2 , for example, it is easy to see that there are eight parallel classesof sectors and they are arranged naturally in the form of a circuit.

For each root α ∈ Φ and each integer k, let Kα,k be the set of alcovesin the half-space

v ∈ V | α · v > −k

(i.e. on one side of the hyperplane Hα,k ). Then Kα,k is a root of ΣΠ andevery root is of this form. Let K∞

α,k be the set of parallel classes of thesectors contained in the root Kα,k . Then K∞

α,k = K∞α,0 for all k, and the

map α → K∞α,0 is a bijection from the root system Φ to the set of roots

in of Σ∞Π (as defined at the end of §1).

Now suppose that ∆ is a building of type Π = X . Then each apart-ment of ∆ is isomorphic to the chamber system ΣΠ , which we can thinkof now as described in terms of alcoves. In particular, each apartmentcontains sectors. A sector of ∆ is a sector in one of its apartments. Wedeclare that two sectors of ∆ (not necessarily in the same apartment)are parallel if their intersection is a sector and we denote by S∞ the par-allel class containing a sector S. Then the set ∆∞ of sector equivalenceclasses has, canonically, the structure of a building of type X called thebuilding at infinity of ∆. Since X is a spherical diagram, ∆∞ is, in fact,a spherical building. For each apartment A of ∆, the set A∞ of parallelclasses S∞ for all sectors S contained in A is an apartment of ∆∞, andthe map A → A∞ is a bijection from the set of all apartments of ∆ tothe set of all apartments of ∆∞.

5 Bruhat-Tits Buildings

We define a Bruhat-Tits building to be an (irreducible thick) affine build-ing of rank at least three whose building at infinity is Moufang (as de-fined in §3). We suppose from now on that ∆ is a Bruhat-Tits buildingof type Π = X and that A is an apartment of ∆. Thus ≥ 2 andby the classification of Moufang spherical buildings summarized in §3,the building at infinity ∆∞ is uniquely determined by the root groups


associated with all the roots of the fixed apartment Σ := A∞ (i.e. by theroot datum of ∆∞ based at Σ), and this root datum is, in turn, uniquelydetermined by a suitable algebraic structure Λ. (Thus Λ is a field or askew-field or an anisotropic quadratic space a Jordan division algebra,etc., according to the family of Moufang spherical buildings to which∆∞ belongs.)

The elements in the root groups of ∆∞ all have unique extensions toautomorphisms of ∆.5 As indicated at the end of §4, we can identify theapartment A of ∆ with the chamber system ΣΠ described in terms of aroot system and alcoves in §4. Thus, in particular, the roots of A are ofthe form Kα,k for (α, k) ∈ Φ × Z (where Kα,k is as defined at the endof §4) and the map α → K∞

α,0 is a bijection from Φ to the set of rootsof Σ. Let a = K∞

α,0 be a root of Σ. Then for each g ∈ U∗a , there exists

a unique k ∈ Z such that the fixed point set of g in A is the root Kα,k .Let ϕa denote the map g → k from U∗

a to Z. We extend ϕa to Ua bysetting ϕa(1) = ∞.

We use additive notation for the group Ua even though Ua is, in somecases, non-abelian. If g, h ∈ Ua , then g + h acts trivially on the set ofalcoves in the half-space

v ∈ V | α · v ≥ −m,

where m = minϕa(g), ϕa(h). Thus

ϕa(g + h) ≥ minϕa(g), ϕb(h) (5.1)

for all roots a of Σ and all g, h ∈ Ua . Since inverses have the same fixedpoint sets, we also have ϕa(−g) = ϕa(g) for each g ∈ U∗

a . It follows thatthe map ∂a from Ua × Ua to R given by

∂a(g, h) = 2−ϕa (g−h) (5.2)

for all (g, h) ∈ Ua × Ua is a metric on Ua . The group Ua must, in fact,be complete with respect to this metric for all roots a of Σ.6

Let

ϕ = ϕa | a ∈ Ξ,

where Ξ denotes the set of roots of the apartment Σ. The inequality(5.1) is called condition (V1) in [8] (on page 139). There are two more

5 The proof of this (in [11]) and the proof (in [4]) of the existence of the building atinfinity ∆∞ are two of the high points in the classification.

6 This is a consequence of the fact that, for the sake of simplicity, we are implicitlyworking with the set of all apartments of ∆ in this article rather than a system ofapartments; see page 129 in [8].


conditions on the set ϕ called (V2) and (V3) on page 139 of [8] whichcan be derived similarly.

Now let Ξ0 and K be as described at the end of §3 and choose a ∈ Ξ0 .Thus the root group Ua is isomorphic to the additive group of K. Asthe inequality (5.1) suggests, ϕa is determined by a discrete valuation ν

of K.7 Since Ua must be complete with respect to the metric ∂a definedin equation (5.2), the field K must be complete with respect to ν . Theexistence of such a valuation ν is thus a necessary condition for ∆ toexist (given the spherical building ∆∞).

Now let ∆0 be an arbitrary spherical building of rank at least twosatisfying the Moufang condition, let Σ be an apartment of ∆0 and letΞ be the set of roots of Σ. A valuation of the root datum of ∆0 based atΣ is a collection of surjective maps ϕa from U∗

a to Z, one for each roota ∈ Ξ, satisfying the three conditions (V1), (V2) and (V3). Note thatwe are now using the word “valuation” in two different ways, to refer toa discrete valuation of a field and to refer to a valuation of a root datum.

We can now formulate a brief version of the classification of Bruhat-Tits buildings:

Theorem. Let ∆0 be a Moufang spherical building of type Π0 definedover K (as defined at the end of §3) and suppose that K is completewith respect to a discrete valuation ν. Let Σ be an apartment of ∆0 , letΞ be the set of roots of Σ and let Ξ0 be as described at the end of §3. 8

Then the following hold:

(i) There is a unique valuation ϕ of the root datum of ∆0 based at Σsuch that ϕa is determined by ν for each a ∈ Ξ0 .

(ii) There is a unique Moufang spherical building ∆0 of type Π0 con-taining ∆0 as a subbuilding and a unique valuation

ϕ = ϕa | a ∈ Ξ

of the root datum of ∆0 based at Σ such that for each root a ∈ Ξ,the root group Ua of ∆0 contains the root group Ua as a subgroup,the map ϕa is the restriction of ϕa to Ua and the root group Ua isthe completion of Ua with respect to the metric given in equation(5.2).

7 A discrete valuation of a field K is a surjective homomorphism ν from the multi-plicative group K∗ to the additive group Z such that ν(u + v) ≥ minν(u), ν(v)for all u, v ∈ K such that u = −v.

8 For the sake of simplicity, we are ignoring the Moufang quadrangles of indiffer-ent and exceptional type (as indicated in the footnote at the end of §3) in theformulation of this result. In fact, similar assertions hold in these cases as well.


(iii) There is a unique affine Coxeter diagram Π with derived diagramΠ0 and a unique Bruhat-Tits building ∆ of type Π such that ∆∞ ∼=∆0 and the valuation ϕ arises from ∆ as described above.

Now suppose that ∆0 is an arbitrary Moufang spherical building withdefining field K. If there exists an affine building ∆ such that ∆∞ ∼= ∆0 ,then, as we observed in §5, K must be complete with respect to a discretevaluation (so the Theorem applies) and, in fact, all the root groups Ua

must be complete with respect to the metric ∂a defined in equation (5.2),so ∆0 = ∆0 .

If ∆0 is the spherical building associated with the F -rational points ofan absolutely simple algebraic group G of F -rank at least two (in whichcase F ⊂ K and K is finite-dimensional over F ) and K is completewith respect to a discrete valuation, then it is always true that ∆0 =∆0 . Furthermore, F is complete with respect to a discrete valuation ifand only if K is, and if either F or K is complete with respect to adiscrete valuation, then the valuation is unique. We conclude that ∆0

is the building at infinity of a Bruhat-Tits building ∆ if and only if F iscomplete with respect to a discrete valuation, and if F is complete withrespect to a discrete valuation, then the Bruhat-Tits building ∆ and itstype Π are uniquely determined by F and G.9

The notion of a valuation of a root datum ϕ makes perfectly good senseif the requirement that “each ϕa maps U∗

a surjectively to Z” is replacedby “each ϕa maps U∗

a to R.” If these images are non-discrete subgroupsof R, then ϕ no longer corresponds to an affine building. In [4] and[11], however, Bruhat and Tits showed that there is a class of geometricstructures which are classified by these “non-discrete” valuations of aroot datum. These structures structures are now usually referred to asnon-discrete buildings although they are not really buildings. See also[6] and [7] for more details about them

Bibliography[1] P. Abramenko and K. S. Brown, Approaches to Buildings, Springer,

Berlin, Heidelberg, New York, 2007.[2] N. Bourbaki, Elements of Mathematics: Lie Groups and Lie Algebras,

Chapters 4–6, Springer, Berlin, Heidelberg, New York, 1968.[3] K. S. Brown, Buildings, Springer, Berlin, Heidelberg, New York, 1989.

9 The standard reference for the connection between algebraic groups over a localfield and affine building is [10].


[4] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, I. Donnéesradicielles valuées, Publ. Math. I.H.E.S. 41 (1972), 5-252.

[5] P. Garrett, Buildings and classical groups, Chapman & Hall, London,1997.

[6] B. Kleiner and Leeb, B., Rigidity of quasi-isometries for symmetric spacesand Euclidean buildings, Inst. Hautes Ëtudes Sci. Publ. Math. 86 (1997),115-197.

[7] A. Parreau, Immeubles affines: construction par les normes et étude desisométries, in: Crystallographic Groups and Their Generalizations (Kor-trijk, 1999), Contemp. Math. 262, Amer. Math. Soc., Providence, 2000,pp. 263-302.

[8] M. A. Ronan, Lectures on Buildings, Academic Press, London, New York,San Diego, 1989.

[9] J. Tits, Buildings of Spherical Type and finite BN-pairs, Lecture Notes inMathematics, vol. 386, Springer, Berlin, Heidelberg, New York, 1974.

[10] J. Tits, Reductive groups over local fields, in Proc. Symp. Pure Math. 33,Part 1 (Automorphic Forms, Representations and L-Functions, Corvallis1977), pp. 29-69, Amer. Math. Soc., 1979.

[11] Tits, J., Immeubles de type affine, in Buildings and the Geometry ofDiagrams (Como 1984), pp. 159-190, Lecture Notes in Mathematics 1181,Springer, New York, Heidelberg, Berlin, 1986.

[12] J. Tits and R. M. Weiss, Moufang Polygons, Springer Monographs inMathematics, Springer, Berlin, Heidelberg, New York, 2002.

[13] R. M. Weiss, The Structure of Spherical Buildings, Princeton UniversityPress, Princeton, 2006.

[14] R. M. Weiss, The Structure of Affine Buildings, Annals of MathematicsStudies, Princeton University Press, to appear.

13Emmy Noether and Hermann Weyl.1

Peter RoquetteMathematisches InstitutUniversität Heidelberg

[email protected]

Contents1 Preface 2852 Introduction 2873 The first period: until 1915 288

3.1 Their mathematical background 2893.2 Meeting in Göttingen 1913 291

4 The second period: 1915-1920 2935 The third period: 1920-1932 295

5.1 Innsbruck 1924 and the method ofabstraction 296

5.2 Representations: 1926/27 2995.3 A letter from N to W: 1927 3025.4 Weyl in Göttingen: 1930-1933 304

6 Göttingen exodus: 1933 3077 Bryn Mawr: 1933-1935 3128 The Weyl-Einstein letter to the NYT 3149 Appendix: documents 318

9.1 Weyl’s testimony 3189.2 Weyl’s funeral speech 3199.3 Letter to the New York Times 3219.4 Letter of Dr. Stauffer-McKee 323

1 Preface

We are here for a conference in honor of Hermann Weyl and so I maybe allowed, before touching the main topic of my talk, to speak aboutmy personal reminiscences of him.

It was in the year 1952. I was 24 and had my first academic job

1 This is the somewhat extended manuscript of a talk presented at the HermannWeyl conference in Bielefeld, September 10, 2006.

285

286 Peter Roquette

at München when I received an invitation from van der Waerden togive a colloquium talk at Zürich University. In the audience of my talkI noted an elder gentleman, apparently quite interested in the topic.Afterwards – it turned out to be Hermann Weyl – he approached meand proposed to meet him next day at a specific point in town. There hetold me that he wished to know more about my doctoral thesis, whichI had completed two years ago already but which had not yet appearedin print. Weyl invited me to join him on a tour on the hills aroundZürich. On this tour, which turned out to last for several hours, I had toexplain to him the content of my thesis which contained a proof of theRiemann hypothesis for function fields over finite base fields. He wasnever satisfied with sketchy explanations, his questions were always tothe point and he demanded every detail. He seemed to be well informedabout recent developments.

This task was not easy for me, without paper and pencil, nor black-board and chalk. So I had a hard time. Moreover the pace set by Weylwas not slow and it was not quite easy to keep up with him, in walkingas well as in talking.

Much later only I became aware of the fact that this tour was a kindof examination, Weyl wishing to find out more about that young manwho was myself. It seems that I did not too bad in this examination,for some time later he sent me an application form for a grant-in-aidfrom the Institute for Advanced Study in Princeton for the academicyear 1954/55. In those years Weyl was commuting between Zürich andPrinceton on a half-year basis. In Princeton he had found, he wroteto me, that there was a group of people who were working in a similardirection.

Hence I owe to Hermann Weyl the opportunity to study in Princeton.The two academic years which I could work and learn there turned outto be important for my later mathematical life. Let me express, posthu-mously, my deep gratitude and appreciation for his help and concern inthis matter.

The above story shows that Weyl, up to his last years, continued to beactive helping young people find their way into mathematics. He reallycared. I did not meet him again in Princeton; he died in 1955.

Let us now turn to the main topic of this talk as announced in thetitle.

Emmy Noether and Hermann Weyl 287

2 Introduction

Both Hermann Weyl and Emmy Noether belonged to the leading groupof mathematicians in the first half of 20th century, who shaped the imageof mathematics as we see it today.

Emmy Noether was born in 1882 in the university town of Erlangen,as the daughter of the renowned mathematician Max Noether. We re-fer to the literature for information on her life and work, foremost tothe empathetic biography by Auguste Dick [8] which has appeared in1970, the 35th year after Noether’s tragic death. It was translated intoEnglish in 1981. For more detailed information see, e.g., the very care-fully documented report by Cordula Tollmien [47]. See also Kimberling’spublications on Emmy Noether, e.g., his article in [5].

When the Nazis had come to power in Germany in 1933, EmmyNoether was dismissed from the University of Göttingen and she emi-grated to the United States. She was invited by Bryn Mawr College as avisiting professor where, however, she stayed and worked for 18 monthsonly, when she died on April 14, 1935 from complications following atumor operation.2

Quite recently we have found the text, hitherto unknown, of thespeech which Hermann Weyl delivered at the funeral ceremony for EmmyNoether on April 17, 1935.3 That moving text puts into evidence thatthere had evolved a close emotional friendship between the two. Therewas more than a feeling of togetherness between immigrants in a newand somewhat unfamiliar environment. And there was more than highesteem for this women colleague who, as Weyl has expressed it4 , was“superior to him in many respects”. This motivated us to try to findout more about their mutual relation, as it had developed through theyears.

We would like to state here already that we have not found manydocuments for this. We have not found letters which they may haveexchanged.5 Neither did Emmy Noether cite Hermann Weyl in her pa-pers nor vice versa6 . After all, their mathematical activities were goinginto somewhat different directions. Emmy Noether’s creative power wasdirected quite generally towards the clarification of mathematical struc-

2 See footnote 49.3 See [37]. We have included in the appendix an English translation of Weyl’s text;

see section 9.2.4 See [53].5 With one exception; see section 5.3.6 There are exceptions; see section 4.

288 Peter Roquette

tures and concepts through abstraction, which means leaving all unnec-essary entities and properties aside and concentrating on the essentials.Her basic work in this direction can be subsumed under algebra, but hermethods eventually penetrated all mathematical fields, including num-ber theory and topology.

On the other side, Hermann Weyl’s mathematical horizon was wide-spread, from complex and real analysis to algebra and number theory,mathematical physics and logic, also continuous groups, integral equa-tions and much more. He was a mathematical generalist in a broadsense, touching also philosophy of science. His mathematical writingshave a definite flair of art and poetry, with his book on symmetry as aculmination point [54].

We see that the mathematical style as well as the extent of Weyl’sresearch work was quite different from that of Noether. And from allwe know the same can be said about their way of living. So, how did itcome about that there developed a closer friendly relationship betweenthem? Although we cannot offer a clear cut answer to this question,I hope that the reader may find something of interest in the followinglines.

3 The first period: until 1915

In the mathematical life of Emmy Noether we can distinguish four peri-ods.7 In her first period she was residing in Erlangen, getting her math-ematical education and working her way into abstract algebra guided byErnst Fischer, and only occasionally visiting Göttingen. The second pe-riod starts in the summer of 1915 when she came to Göttingen for good,in order to work with Klein and Hilbert. This period is counted untilabout 1920. Thereafter there begins her third period, when her famouspaper “Idealtheorie in Ringbereichen” (Ideal theory in rings) appeared,with which she “embarked on her own completely original mathematicalpath” – to cite a passage from Alexandroff’s memorial address [2]. Thefourth period starts from 1933 when she was forced to emigrate and wentto Bryn Mawr.

7 Weyl [53] distinguishes three epochs but they represent different time intervalsthan our periods.


3.1 Their mathematical background

Hermann Weyl, born in 1885, was about three years younger than EmmyNoether. In 1905, when he was 19, he entered Göttingen University (af-ter one semester in München). On May 8, 1908 he obtained his doctoratewith a thesis on integral equations, supervised by Hilbert.

At about the same time (more precisely: on December 13, 1907)Emmy Noether obtained her doctorate from the University of Erlan-gen, with a thesis on invariants supervised by Gordan. Since she wasolder than Weyl we see that her way to Ph.D. was longer than his. Thisreflects the fact that higher education, at that time, was not as open tofemales as it is today; if a girl wished to study at university and get aPh.D. then she had to overcome quite a number of difficulties arisingfrom tradition, prejudice and bureaucracy. Noether’s situation is welldescribed in Tollmien’s article [47].8

But there was another difference between the status of Emmy Noetherand Hermann Weyl at the time of their getting the doctorate.

On the one side, Weyl was living and working in the unique Göttin-gen mathematical environment of those years. Weyl’s thesis belongs tothe theory of integral equations, the topic which stood in the center ofHilbert’s work at the time, and which would become one of the sourcesof the notion of “Hilbert space”. And Weyl’s mathematical curiosity wasnot restricted to integral equations. In his own words, he was captivatedby all of Hilbert’s mathematics. Later he wrote:9

I resolved to study whatever this man [Hilbert] had written. At the end of myfirst year I went home with the “Zahlbericht” under my arm, and during thesummer vacation I worked my way through it - without any previous knowledgeof elementary number theory or Galois theory. These were the happiest monthsof my life, whose shine, across years burdened with our common share of doubtand failure, still comforts my soul.

We see that Weyl in Göttingen was exposed to and responded to thenew and exciting ideas which were sprouting in the mathematical worldat the time. His mathematical education was strongly influenced by hisadvisor Hilbert.

On the other side, Noether lived in the small and quiet mathematicalworld of Erlangen. Her thesis, supervised by Paul Gordan, belongs toclassical invariant theory, in the framework of so-called symbolic compu-tations. Certainly this did no longer belong to the main problems which

8 For additional material see also Tollmien’s web page: www.tollmien.com.9 Cited from the Weyl article in “MacTutor History of Mathematics Archive”.

290 Peter Roquette

dominated mathematical research in the beginning of the 20th century.It is a well-known story that after Hilbert in 1888 had proved the finite-ness theorem of invariant theory which Gordan had unsuccessfully triedfor a long time, then Gordan did not accept Hilbert’s existence proofsince that was not constructive in his (Gordan’s) sense. He declaredthat Hilbert’s proof was “theology, not mathematics”. Emmy Noether’swork was fully integrated into Gordan’s formalism and so, in this way,she was not coming near to the new mathematical ideas of the time.10

In later years she described the work of her thesis as rubbish (“Mist” inGerman11). In a letter of April 14, 1932 to Hasse she wrote:

Ich habe das symbolische Rechnen mit Stumpf und Stil verlernt.I have completely forgotten the symbolic calculus.

We do not know when Noether had first felt the desire to update hermathematical background. Maybe the discussions with her father helpedto find her way; he corresponded with Felix Klein in Göttingen and sowas well informed about the mathematical news from there. She herselfreports that it was mainly Ernst Fischer who introduced her to whatwas then considered “modern” mathematics. Fischer came to Erlangenin 1911, as the successor of the retired Gordan.12 In her curriculum vitaewhich she submitted in 1919 to the Göttingen Faculty on the occasionof her Habilitation, Noether wrote:

Wissenschaftliche Anregung verdanke ich wesentlich dem persönlichen mathema-tischen Verkehr in Erlangen und in Göttingen. Vor allem bin ich Herrn E. Fischerzu Dank verpflichtet, der mir den entscheidenden Anstoß zu der Beschäftigung mitabstrakter Algebra in arithmetischer Auffassung gab, was für all meine späterenArbeiten bestimmend blieb.

I obtained scientific guidance and stimulation mainly through personal math-ematical contacts in Erlangen and in Göttingen. Above all I am indebted toMr. E. Fischer from whom I received the decisive impulse to study abstractalgebra from an arithmetical viewpoint, and this remained the governing ideafor all my later work.

Thus it was Fischer under whose direction Emmy Noether’s mathe-

10 Well, Noether had studied one semester in Göttingen, winter 1903/04. But shefell ill during that time and had to return to her home in Erlangen, as Tollmien[47] reports. We did not find any indication that this particular semester has hada decisive influence on her mathematical education.

11 Cited from Auguste Dick’s Noether biography [8].12 More precisely: Gordan retired in 1910 and was followed by Erhard Schmidt who,

however, left Erlangen one year later already and was followed in turn by ErnstFischer. – The name of Fischer is known from the Fischer-Riesz theorem infunctional analysis.


matical outlook underwent the “transition from Gordan’s formal stand-point to the Hilbert method of approach”, as Weyl stated in [53].

We may assume that Emmy Noether studied, like Weyl, all of Hilbert’spapers, at least those which were concerned with algebra or arithmetic.In particular she would have read the paper [14] where Hilbert provedthat every ideal in a polynomial ring is finitely generated; in her famouslater paper [29] she considered arbitrary rings with this property, whichtoday are called “Noetherian rings”. We may also assume that Hilbert’sZahlbericht too was among the papers which Emmy Noether studied; itwas the standard text which every young mathematician of that timeread if he/she wished to learn algebraic number theory. We know from alater statement that she was well acquainted with it – although at thatlater time she rated it rather critically13 , in contrast to Weyl who, aswe have seen above, was enthusiastic about it. But not only Hilbert’spapers were on her agenda; certainly she read Steinitz’ great paper “Al-gebraische Theorie der Körper” [44] which marks the start of abstractfield theory. This paper is often mentioned in her later publications, asthe basis for her abstract viewpoint of algebra.

3.2 Meeting in Göttingen 1913

Hermann Weyl says in [53], referring to the year 1913:

. . . She must have been to Göttingen about that time, too, but I suppose onlyon a visit with her brother Fritz. At least I remember him much better thanher from my time as a Göttinger Privatdozent, 1910-1913.

We may conclude that he had met Emmy Noether in Göttingen about1913, but also that she did not leave a lasting impression on him on thatoccasion.

As Tollmien [47] reports, it was indeed 1913 when Emmy Noether vis-ited Göttingen for a longer time (together with her father Max Noether).Although we have no direct confirmation we may well assume that shemet Weyl during this time. In the summer semester 1913 Weyl gave twotalks in the Göttinger Mathematische Gesellschaft. In one session he re-ported on his new book “Die Idee der Riemannschen Fläche” (The ideaof the Riemann surface) [51], and in another he presented his proof onthe equidistribution of point sequences modulo 1 in arbitrary dimensions

13 In a letter of November 17, 1926 to Hasse; see [22]. Olga Taussky-Todd [45]reports from later time in Bryn Mawr, that once “Emmy burst out against theZahlbericht, quoting also Artin as having said that it delayed the development ofalgebraic number theory by decades”.

292 Peter Roquette

[52] – both pieces of work have received the status of a “classic” by now.Certainly, Max Noether as a friend of Klein will have been invited tothe sessions of the Mathematische Gesellschaft, and his daughter Emmywith him. Before and after the session people would gather for discus-sion, and from all we know about Emmy Noether she would not havehesitated to participate in the discussions. From what we have said inthe foregoing section we can conclude that her mathematical status wasup-to-date and well comparable to his, at least with respect to algebraand number theory.

Unfortunately we do not know anything about the possible subjectsof the discussions of Emmy Noether with Weyl. It is intriguing to thinkthat they could have talked about Weyl’s new book “The Idea of theRiemann surface”. Weyl in his book defines a Riemann surface ax-iomatically by structural properties, namely as a connected manifold X

with a complex 1-dimensional structure. This was a completely newapproach, a structural viewpoint. Noether in her later period used toemphasize on every occasion the structural viewpoint. The structure inWeyl’s book is an analytic one, and he constructs an algebraic structurefrom this, namely the field of meromorphic funtions, using the so-calledDirichlet principle – whereas Emmy Noether in her later papers alwaysstarts from the function field as an algebraic structure. See, e.g., herreport [27]. There she did not cite Weyl’s book but, of course, this doesnot mean that she did not know it.

We observe that the starting idea in Weyl’s book was the definitionand use of an axiomatically defined topological space14 . We wonderwhether this book was the first instance where Emmy Noether was con-fronted with the axioms of what later was called a topological space.It is not without reason to speculate that her interest in topology wasinspired by Weyl’s book. In any case, from her later cooperation withPaul Alexandroff we know that she was acquainted with problems oftopology; her contribution to algebraic topology was the notion of “Bettigroup” instead of the “Betti number” which was used before. Let us citeAlexandroff in his autobiography [1]:

In the middle of December Emmy Noether came to spend a month in Blaricum.This was a brilliant addition to the group of mathematicians around Brouwer.I remember a dinner at Brouwer’s in her honour during which she explainedthe definition of the Betti groups of complexes, which spread around quicklyand completely transformed the whole of topology.

14 The Hausdorff axiom was not present in the first edition. This gap was filled inlater editions.


This refers to December 1925. Blaricum was the place where L. E. J.Brouwer lived.

We have mentioned this contact of Emmy Noether to the group aroundBrouwer since Weyl too did have mathematical contact with Brouwer.In fact, in his book “The idea of the Riemann surface” Weyl mentionedBrouwer as a source of inspiration. He writes:

In viel höherem Maße, als aus den Zitaten hervorgeht, bin ich dabei durch diein den letzten Jahren erschienenen grundlegenden topologischen UntersuchungenBrouwers, dessen gedankliche Schärfe und Konzentration man bewundern muss,gefördert worden . . .

I have been stimulated – much more than the citations indicate – by therecent basic topological investigations of Brouwer, whose ideas have to be ad-mired in their sharpness and concentration.

Brouwer’s biographer van Dalen reports that Weyl and Brouwer met sev-eral times in the early 1920s [49]. By the way, Emmy Noether, HermannWeyl and L. E. J. Brouwer met in September 1920 in Bad Nauheim, atthe meeting of the DMV.15

Returning to the year 1913 in Göttingen: In the session of July 30,1913 of the Göttinger Mathematische Gesellschaft, Th. v. Kármán re-ported on problems connected with a recent paper on turbulence byEmmy Noether’s brother Fritz. Perhaps Fritz too was present in Göt-tingen on this occasion, and maybe this was the incident why Weyl hadremembered not only Emmy but also Fritz? That he remembered Fritz“much better” may be explained by the topic of Fritz’ paper; questions ofturbulence lead to problems about partial differential equations, whichwas at that time more close to Weyl’s interests than were algebraic prob-lems which Emmy pursued.

4 The second period: 1915-1920

In these years Emmy Noether completed several papers which are ofalgebraic nature, mostly about invariants, inspired by the Göttingenmathematical atmosphere dominated by Hilbert. She also wrote a re-port in the Jahresbericht der DMV on algebraic function fields, in whichNoether compares the various viewpoints of the theory: analytic, geo-metric and algebraic (which she called “arithmetic”) and she points outthe analogies to the theory of number fields. That was quite well knownto the people working with algebraic functions, but perhaps not writ-ten up systematically as Emmy Noether did. Generally, these papers of

15 DMV = Deutsche Mathematiker Vereinigung = German Mathematical Society.

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hers can be rated as good work, considering the state of mathematicsof the time, but not as outstanding. It is unlikely that Hermann Weylwas particularly interested in these papers; perhaps he didn’t even knowabout them.16

But this would change completely with the appearance of Noether’spaper on invariant variation problems [26] (“Invariante Variationsprob-leme”). The main result of this paper is of fundamental importance inmany branches of theoretical physics even today. It shows a connectionbetween conservation laws in physics and the symmetries of the theory.It is probably the most cited paper of Emmy Noether up to the presentday. In 1971 an English translation appeared [46], and in 2004 a Frenchtranslation with many comments [21].

In 1918, when the paper appeared, its main importance was seen in itsapplicability in the framework of Einstein’s relativity theory. Einsteinwrote to Hilbert in a letter of May 24, 1918:

Gestern erhielt ich von Frl. Noether eine sehr interessante Arbeit über Invarianten-bildung. Es imponiert mir, dass man diese Dinge von so allgemeinem Standpunktübersehen kann . . . Sie scheint ihr Handwerk zu verstehen.

Yesterday I received from Miss Noether a very interesting paper on the for-mation of invariants. I am impressed that one can handle those things fromsuch a general viewpoint . . . She seems to understand her job.

Einstein had probably met Emmy Noether already in 1915 during hisvisit to Göttingen.

Emmy Noether’s result was the fruit of a close cooperation withHilbert and with Klein in Göttingen during the past years. As Weyl[53] reports, “Hilbert at that time was over head and ears in the generaltheory of relativity, and for Klein, too, the theory of relativity broughtthe last flareup of his mathematical interest and production”. EmmyNoether, although she was doubtless influenced, not only assisted thembut her work was a genuine production of her own. In particular, theconnection of invariants with the symmetry groups, with its obvious ref-erence to Klein’s Erlanger program, caught the attention of the worldof mathematicians and theoretical physicists.17 Noether’s work in thisdirection has been described in detail in, e.g., [39], [21], [57].

It is inconceivable that Hermann Weyl did not take notice of thisimportant work of Emmy Noether. At that time Weyl, who was in cor-

16 In those years Weyl was no more in Göttingen but held a professorship at ETH inZürich.

17 In [39] it is said that nevertheless “few mathematicians and even fewer physisistsever read Noether’s original article . . . ”.


respondence with Hilbert and Einstein, was also actively interested inthe theory of relativity; his famous book “Raum, Zeit, Materie” (Space,Time, Matter) had just appeared. Emmy Noether had cited Weyl’sbook18 , and almost certainly she had sent him a reprint of her paper.Thus, through the medium of relativity theory there arose mathematicalcontact between them.19 Although we do not know, it is well conceivablethat there was an exchange of letters concerning the mathematical the-ory of relativity. From now on Weyl would never remember her brotherFritz better than Emmy.

In 1919 Emmy Noether finally got her Habilitation. Already in 1915Hilbert and Klein, convinced of her outstanding qualification, had rec-ommended her to apply for Habilitation. She did so, but it is a sadstory that it was unsuccessful because of her gender although her sci-entific standing was considered sufficient. The incident is told in detailin Tollmien’s paper [47]. Thus her Habilitation was delayed until 1919after the political and social conditions had changed.

We see again the difference between the scientific careers of Weyl andof Emmy Noether. Weyl had his Habilitation already in 1910, and since1913 he held a professorship in Zürich. Emmy Noether’s Habilitationwas possible only nine years later than Weyl’s. As is well known, shenever in her life got a permanent position; although in the course of timeshe rose to become one of the leading mathematicians in the world.

5 The third period: 1920-1932

The third period of Noether’s mathematical life starts with the greatpaper “Idealtheorie in Ringbereichen” (Ideal theory in rings) [29].20 AfterHilbert had shown in 1890 that in a polynomial ring (over a field asbase) every ideal is finitely generated, Noether now takes this propertyas an axiom and investigates the primary decomposition of ideals inarbitrary rings satisfying this axiom. And she reformulates this axiomas an “ascending chain condition” for ideals. Nowadays such rings arecalled “Noetherian”. The paper appeared in 1921.

We note that she was nearly 40 years old at that time. The mathemat-

18 The citation is somewhat indirect. Noether referred to the literature cited in apaper by Felix Klein [20], and there we find Weyl’s book meontioned. In a secondpaper of Noether [30] Weyl’s book is cited directly.

19 Added in proof: We read in [39] that there is a reference to Noether in Weyl’sbook, tucked away in a footnote.

20 Sometimes the earlier investigation jointly with Schmeidler [28] is also counted asbelonging to this period.

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ical life of Emmy Noether is one of the counterexamples to the dictumthat mathematics is a science for the young and the most creative workis done before 40. Emmy Noether would not have been a candidate forthe Fields medal if it had already existed at that time.

5.1 Innsbruck 1924 and the method of abstraction

We do not know whether and how Weyl took notice of the above-mentioned paper of Noether [29]. But her next great result, namely thefollow-up paper [32] on the ideal theory of what are now called Dedekindrings, was duly appreciated by Weyl. At the annual DMV meeting in1924 in Innsbruck Noether reported about it [31]. And Weyl was chair-ing that session; so we know that he was informed first hand about herfundamental results.

In her talk, Emmy Noether defined Dedekind rings by axioms andshowed that every ring satisfying those axioms admits a unique fac-torization of ideals into prime ideals. Well, Noether did not use theterminology “Dedekind ring”; this name was coined later. Instead, sheused the name “5-axioms-ring” since in her enumeration there were 5axioms. Then she proved that the ring of integers in a number fieldsatisfies those axioms, and similarly in the funtion field case. This is agood example of Noether’s “method of abstraction”. By working solelywith those axioms she first generalized the problem, and it turned outthat by working in this generalization the proof of prime decompositionis simplified if compared with the former proofs (two of which had beengiven by Hilbert [15]).

How did Weyl react to Noether’s method of abstraction? At thattime, this method met sometimes with skepticism and even rejectionby mathematicians. But Hilbert in various situations had already takenfirst steps in this direction and so Weyl, having been Hilbert’s doctorand,was not against Noether’s method. After all, in his book “Space, Time,Matter” Weyl had introduced vector spaces by axioms, not as n-tuples21 .

Weyl’s reaction can be extracted implicitly from an exchange of letterswith Hasse which happened seven years later. The letter of Weyl is datedDecember 8, 1931. At that time Weyl held a professorship in Göttingen(since 1930) as the successor of Hilbert. Thus Emmy Noether was nowhis colleague in Göttingen. Hasse at that time held a professorship inMarburg (also since 1930) as the successor of Hensel. The occasion of

21 This has been expressly remarked by MacLane [23].


Weyl’s letter was the theorem that every simple algebra over a numberfield is cyclic; this had been established some weeks ago by Brauer,Hasse and Noether, and the latter had informed Weyl about it. So Weylcongratulated Hasse for this splendid achievement. And he recalled themeeting in Innsbruck 1924 when he first had met Hasse.

For us, Hasse’s reply to Weyl’s letter is of interest.22 Hasse answeredon December 15, 1931. First he thanked Weyl for his congratulations,but at the same time pointed out that the success was very essentiallydue also to the elegant theory of Emmy Noether, as well as the p-adictheory of Hensel. He also mentioned Minkowski in whose work the ideaof the Local-Global principle was brought to light very clearly. And thenHasse continued, recalling Innsbruck:

Auch ich erinnere mich sehr gut an Ihre ersten Worte zu mir anläßlich meinesVortrages über die erste explizite Reziprozitätsformel für höheren Exponenten inInnsbruck. Sie zweifelten damals ein wenig an der inneren Berechtigung solcherUntersuchungen, indem Sie ins Feld führten, es sei doch gerade Hilberts Verdi-enst, die Theorie des Reziprozitätsgesetzes von den expliziten Rechnungen frühererForscher, insbesondere Kummers, befreit zu haben.

I too remember very well your first words to me on the occasion of my talk inInnsbruck, about the first explicit reciprocity formula for higher exponent. Yousomewhat doubted the inner justification of such investigations, by pointingout that Hilbert had freed the theory of the reciprocity law from the explicitcomputations of former mathematicians, in particular Kummer’s.

We conclude: Hasse in Innsbruck had talked on explicit formulas andWeyl had critized this, pointing out that Hilbert had embedded thereciprocity laws into more structural results. Probably Weyl had inmind the product formula for the so-called Hilbert symbol which, in asense, comprises all explicit reciprocity formulas.23 For many, like Weyl,this product formula was the final word on reciprocity while for Hassethis was the starting point for deriving explicit, constructive reciprocityformulas, using heavily the p-adic methods of Hensel.

We can fairly well reconstruct the situation in Innsbruck: EmmyNoether’s talk had been very abstract, and Hasse’s achievement wasin some sense the opposite since he was bent on explicit formulas, andquite involved ones too.24 Weyl had been impressed by Emmy Noether’s

22 We have found Hasse’s letter in the Weyl legacy in the archive of the ETH inZürich.

23 But Hilbert was not yet able to establish his product formula in full generality. Werefer to the beautiful and complete treatment in Hasse’s class field report, Part 2[10] which also contains the most significant historic references.

24 Hasse’s Innsbruck talk is published in [9]. The details are found in volume 154 ofCrelle’s Journal where Hasse had published 5 papers on explicit reciprocity laws.

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achievements which he considered as continuing along the lines set byHilbert’s early papers on number theory. In contrast, he consideredHasse’s work as pointing not to the future but to the mathematicalpast.

We have mentioned here these letters Weyl-Hasse in order to put intoevidence that already in 1924, Weyl must have had a very positive opin-ion on Emmy Noether’s methods, even to the point of preferring it toexplicit formulas.

But as it turned out, Hasse too had been impressed by Noether’slecture. In the course of the years after 1924, as witnessed by the Hasse-Noether correspondence [22], Hasse became more and more convincedabout Noether’s abstract methods which, in his opinion, served to clarifythe situation; he used the word “durchsichtig” (lucid). Hasse’s addressat the DMV meeting in Prague 1929 [11] expresses his views very clearly.Hensel’s p-adic methods could also be put on an abstract base, due tothe advances in the theory of valuations.25 But on the other hand, Hassewas never satisfied with abstract theorems only. In his cited letter toWeyl 1931 he referred to his (Hasse’s) class field report Part II [10] whichhad appeared just one year earlier. There, he had put Artin’s generalreciprocity law26 as the base, and from this structural theorem he wasable to derive all the known reciprocity formulas. Hasse closed his letterwith the following:

. . . Ich kann aber natürlich gut verstehen, daß Dinge wie diese expliziten Reziproz-itätsformeln einem Manne Ihrer hohen Geistes- und Geschmacksrichtung wenigerzusagen, als mir, der ich durch die abstrakte Mathematik Dedekind-E. NoetherscherArt nie restlos befriedigt bin, ehe ich nicht zum mindesten auch eine explizite,formelmäßige konstruktive Behandlung daneben halten kann. Erst von der let-zteren können sich die eleganten Methoden und schönen Ideen der ersteren wirk-lich vorteilhaft abheben.

. . . But of course I well realize that those explicit reciprocity formulas maybe less attractive to a man like you with your high mental powers and taste, asto myself. I am never fully satisfied by the abstract mathematics of Dedekind-E. Noether type before I can also supplement it by at least one explicit, com-putational and constructive treatment. It is only in comparison with the latterthat the elegant methods and beautiful ideas of the former can be appreciatedadvantageously.

Here, Hasse touches a problem which always comes up when, as Emmy

25 For this see [36].26 By the way, this was the first treatment of Artin’s reciprocity law in book form

after Artin’s original paper 1927. We refer to our forthcoming book on the Artin-Hasse correspondence.


Noether propagated, the abstract methods are put into the foreground.Namely, abstraction and axiomatization is not to be considered as anend in itself; it is a method to deal with concrete problems of substance.But Hasse was wrong when he supposed that Weyl did not see that prob-lem. Even in 1931, the same year as the above cited letters, Weyl gavea talk on abstract algebra and topology as two ways of mathematicalcomprehension [55]. In this talk Weyl stressed the fact that axiomati-zation is not only a way of securing the logical truth of mathematicalresults, but that it had become a powerful tool of concrete mathematicalresearch itself, in particular under the influence of Emmy Noether. Buthe also said that abstraction and generalization do not make sense with-out mathematical substance behind it. This is close to Hasse’s opinionas expressed in his letter above.27 The mathematical work of both Weyland Hasse puts their opinions into evidence.

At the same conference [55] Weyl also said that the “fertility of theseabstracting methods is approaching exhaustion”. This, however, met withsharp protests by Emmy Noether, as Weyl reports in [53]. In fact, to-day most of us would agree with Noether. The method of abstractingand axiomatizing has become a natural and powerful tool for the math-ematician, with striking successes until today. In Weyl’s letter to Hasse(which we have not cited fully) there are passages which seem to indicatethat in principle he (Weyl) too would agree with Emmy Noether. For,he encourages Hasse to continue his work in the same fashion, and thereis no mention of an impending “exhaustion”. He closes his letter withthe following sentence which, in our opinion, shows his (Weyl’s) opinionof how to work in mathematics:

Es freut mich besonders, daß bei Ihnen die in Einzelleistungen sich bewährendewissenschaftliche Durchschlagskraft sich mit geistigem Weitblick paart, der überdas eigene Fach hinausgeht.

In particular I am glad that your scientific power, tested in various specialaccomplishments, goes along with a broad view stretching beyond your ownspecial field.

5.2 Representations: 1926/27

We have made a great leap from 1924 to 1931. Now let us return andproceed along the course of time. In the winter semester 1926/27 Her-

27 Even more clearly Hasse has expressed his view in the foreword to his beautifuland significant book on abelian fields [13].

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mann Weyl stayed in Göttingen as a visiting professor, and he lecturedon representations of continuous groups. In [53] he reports:

I have a vivid recollection of her [Emmy Noether]. . . She was in the audience;for just at that time the hypercomplex number systems and their representa-tions had caught her interest and I remember many discussions when I walkedhome after the lectures, with her and von Neumann, who was in Göttingen asa Rockefeller Fellow, through the cold, dirty, rain-wet streets of Göttingen.

This gives us information not only about the weather conditions in Göt-tingen in winter time but also that a lively discussion between Weyl andEmmy Noether had developed.

We do not know precisely when Emmy Noether first had become in-terested in the representation theory of groups and algebras, or “hyper-complex systems” in her terminology. In any case, during the wintersemester 1924/25 in Göttingen she had given a course on the subject.And in September 1925 she had talked at the annual meeting of theDMV in Danzig on “Group characters and ideal theory”. There she ad-vocated that the whole representation theory of groups should be sub-sumed under the theory of algebras and their ideals. She showed how theWedderburn theorems for algebras are to be interpreted in representa-tion theory, and that the whole theory of Frobenius on group charactersis subsumed in this way. Although she announced a more detailed pre-sentation in the Mathematische Annalen, the mathematical public hadto wait until 1929 for the actual publication [33]28 . Noether was nota quick writer; she developed her ideas again and again in discussions,mostly on her walks with students and colleagues into the woods aroundGöttingen, and in her lectures.

The text of her paper [33] consists essentially of the notes taken by vander Waerden at her lecture in the winter semester 1927/28. Althoughthe main motivation of Noether was the treatment of Frobenius’ theoryof representations of finite groups, it turned out that finite groups aretreated on the last two pages only – out of a total of 52 pages. The mainpart of the paper is devoted to introducing and investigating generalabstract notions, capable of dealing not only with the classical theoryof finite group representations but with much more. Again we see thepower of Noether’s abstracting methods. The paper has been said toconstitute “one of the pillars of modern linear algebra”.29

28 This appeared in the “Mathematische Zeitschrift” and not in the “Annalen” asannounced by Noether in Danzig.

29 Cited from [7] who in turn refers to Bourbaki.


We can imagine Emmy Noether in her discussions with Weyl on thecold, wet streets in Göttingen 1926/27, explaining to him the essen-tial ideas which were to become the foundation of her results in herforthcoming paper [33]. We do not know to which extent these ideas en-tered Weyl’s book [56] on classical groups. After all, the classical groupswhich are treated in Weyl’s book are infinite while Noether’s theoryaimed at the representation of finite groups. Accordingly, in Noether’swork there appeared a finiteness condition for the algebras considered,namely the descending chain condition for (right) ideals. If one wishesto use Noether’s results for infinite groups one first has to generalize hertheory such as to remain valid in more general cases too. Such a gener-alization did not appear until 1945; it was authored by Nathan Jacobson[16]. He generalized Noether’s theory to simple algebras containing atleast one irreducible right ideal.

At this point let me tell a story which I witnessed in 1947. I wasa young student in Hamburg then. In one of the colloquium talks thespeaker was F. K. Schmidt who recently had returned from a visit to theUS, and he reported on a new paper by Jacobson which he had discoveredthere.30 This was the above mentioned paper [16]. F. K. Schmidt was abrilliant lecturer and the audience was duly impressed. In the ensuingdiscussion Ernst Witt, who was in the audience, commented that all thishad essentially been known to Emmy Noether already.

Witt did not elaborate on his comment. But he had been one of the“Noether boys” in 1932/33, and so he had frequently met her. There isno reason to doubt his statement. It may well have been that she hadtold him, and perhaps others too, that her theory could be generalized inthe sense which later had been found by Jacobson. Maybe she had justgiven a hint in this direction, without details, as was her usual custom.In fact, reading Noether’s paper [33] the generalization is obvious to anyreader who is looking for it.31 It is fascinating to think that the idea forsuch a generalization arose from her discussions with Weyl in Göttingenin 1927, when infinite groups were discussed and the need to generalizeher theory became apparent.

By the way, Jacobson and Emmy Noether met in 1934 in Princeton,when she was running a weekly seminar. We cannot exclude the possi-

30 In those post-war years it was not easy to get hold of books or journals fromforeign countries, and so the 1945 volume of the “Transactions” was not availableat the Hamburg library.

31 A particularly short and beautiful presentation is to be found in Artin’s article [4]where he refers to Tate.

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bility that she had given a hint to him too, either in her seminar or inpersonal discussions. After all, this was her usual style, as reported byvan der Waerden [48].

5.3 A letter from N to W: 1927

As stated in the introduction we have not found letters from EmmyNoether to Weyl, with one exception. That exception is kept in thearchive of the ETH in Zürich. It is written by Emmy Noether and datedMarch 12, 1927. This is shortly after the end of the winter semester1926/27 when Weyl had been in Göttingen as reported in the previoussection. Now Weyl was back in Zürich and they had to write lettersinstead of just talking.

The letter concerns Paul Alexandroff and Heinz Hopf and their planto visit Princeton in the academic year 1927/28.

We have already mentioned Alexandroff in section 3.2 in connectionwith Noether’s contributions to topology. From 1924 to 1932 he spentevery summer in Göttingen, and there developed a kind of friendly re-lationship between him and Emmy Noether. The relation of Noether toher “Noether boys” has been described by André Weil as like a motherhen to her fledglings [50]. Thus Paul Alexandroff was accepted by EmmyNoether as one of her fledglings. In the summer semester 1926 HeinzHopf arrived in Göttingen as a postdoc from Berlin and he too was ac-cepted as a fledgling. Both Alexandroff and Hopf became close friendsand they decided to try to go to Princeton University in the academicyear 1927/28.

Perhaps Emmy Noether had suggested this; in any case she helpedthem to obtain a Rockefeller grant for this purpose. It seems that Weylalso had lent a helping hand, for in her letter to him she wrote:

. . . Jedenfalls danke ich Ihnen sehr für Ihre Bemühungen; auch Alexandroff undHopf werden Ihnen sehr dankbar sein und es scheint mir sicher, dass wenn dieformalen Schwierigkeiten erst einmal überwunden sind, Ihr Brief dann von we-sentlichem Einfluss sein wird.

. . . In any case I would like to thank you for your help; Alexandroff and Hopftoo will be very grateful to you. And I am sure that if the formal obstacles willbe overcome then your letter will be of essential influence.

The “formal obstacles” which Noether mentioned were, firstly, the factthat originally the applicants (Alexandroff and Hopf) wished to stayfor a period less than an academic year in Princeton (which later theyextended to a full academic year), and secondly, that Hopf’s knowledge


of the English language seemed not to be sufficient in the eyes of theRockefeller Foundation (but Noether assured them that Hopf wantedto learn English).32 But she mentioned there had been letters sent toLefschetz and Birkhoff and that at least the latter had promised toapproach the Rockefeller Foundation to make an exception.

Alexandroff was in Moscow and Hopf in Berlin at the time, and so themother hen acted as representative of her two chickens.33

About Alexandroff’s and Hopf’s year in Princeton we read in theAlexandroff article of MacTutor’s History of Mathematics archive:

Aleksandrov and Hopf spent the academic year 1927-28 at Princeton in theUnited States. This was an important year in the development of topologywith Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz,Veblen and Alexander.

The letter from Noether to Weyl shows that both N. and W. wereinstrumental in arranging this important Princeton year for Alexandroffand Hopf. Both were always ready to help young mathematicians tofind their way.

Remark: Later in 1931, when Weyl had left Zürich for Göttingen,it was Heinz Hopf who succeeded Weyl in the ETH Zürich. At thosetimes it was not uncommon that the leaving professor would be askedfor nominations if the faculty wished to continue his line. We can wellimagine that Weyl, who originally would have preferred Artin34 , finallynominated Heinz Hopf for this position. If so then he would have dis-cussed it with Emmy Noether since she knew Hopf quite well. It mayeven have been that she had taken the initiative and proposed to Weylthe nomination of Hopf. In fact, in the case of Alexandroff she did soin a letter to Hasse dated October 7, 1929 when it was clear that Hassewould change from Halle to Marburg. There she asked Hasse whetherhe would propose the name of Alexandroff as a candidate in Halle.35

It seems realistic to assume that in the case of Heinz Hopf she actedsimilarly.

32 It seems that his knowledge of English had improved in the course of time sinceHeinz Hopf had been elected president of the IMU (International MathematicalUnion) in 1955 till 1958.

33 Probably the letters from Noether and Weyl to the Rockefeller Foundation, i.e., toTrowbridge, on this matter are preserved in the Rockefeller archives in New Yorkbut we have not checked this.

34 In fact, in 1930 Artin received an offer from the ETH Zürich which, however, hefinally rejected.

35 This however, did not work out. The successor of Hasse in Halle was HeinrichBrandt, known for the introduction of “Brandt’s gruppoid” for divisor classes insimple algebras over number fields.

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Remark 2: The above mentioned letter of Noether to Weyl containsa postscript which gives us a glimpse of the mathematical discussionbetween the two (and it is the only written document for this). It reads:

Die Mertens-Arbeit, von der ich Ihnen sprach, steht Monatshefte, Bd. 4. Ichdachte an den Schluss, Seite 329. Es handelt sich hier aber doch nur um De-terminanten-Relationen, sodass es für Sie wohl kaum in Betracht kommt.

The Mertens paper which I mentioned to you is contained in volume 4 ofthe Monatshefte. I had in mind the end of the paper, page 329. But this isconcerned with determinant relations only, hence it will perhaps not be relevantto your purpose.

The Mertens paper is [25]. We have checked the cited page but did notfind any hint which would connect to Weyl’s work. Perhaps someoneelse will be able to interpret Noether’s remark.

5.4 Weyl in Göttingen: 1930-1933

Weyl in [53] reports:

When I was called permanently to Göttingen in 1930, I earnestly tried to obtainfrom the Ministerium a better position for her [Emmy Noether], because Iwas ashamed to occupy such a preferred position beside her whom I knew mysuperior as a mathematician in many respects.

We see that by now, Weyl was completely convinced about the mathe-matical stature of Emmy Noether. After all, Emmy Noether in 1930 wasthe world-wide acknowledged leader of abstract algebra, and her pres-ence in Göttingen was the main attraction for young mathematiciansfrom all over the world to visit the Mathematical Institute and studywith her.

It would be interesting to try to find out which “better position” Weylhad in mind in his negotiations with the Ministerium in Berlin. Maybehe wished tenure for her, and an increase of her salary. The archives inBerlin will perhaps have the papers and reports of Weyl’s negotiations.From those papers one may be able to extract the reasons for the rejec-tion. But the opposition against Noether’s promotion did not only comefrom the Ministerium in Berlin. It seems that a strong opposition camealso from among the mathematician colleagues in Göttingen, for Weylcontinues with his report as follows:

. . . nor did an attempt [succeed] to push through her election as a member ofthe Göttinger Gesellschaft der Wissenschaften. Tradition, prejudice, externalconsiderations, weighted the balance against her scientific merits and scientificgreatness, by that time denied by no one.


I do not know whether there exist minutes of the meetings of the Göt-tinger Mathematische Gesellschaft in 1930. If so then it would be in-teresting to know the traditional, biased and external arguments whichWeyl said were put forward against Emmy Noether from the members ofthe Mathematische Gesellschaft . Was it still mainly her gender? Or wasit the opposition to her “abstract” mathematical methods? In any case,the decision not to admit Emmy Noether as a member of the GöttingerMathematische Gesellschaft is to be regarded as an injustice to her anda lack of understanding of the development of modern mathematics. Af-ter all, the Emmy Noether of 1930 was quite different from the EmmyNoether of 1915. Now in 1930, she had already gone a long way “on herown completely original mathematical path” , and her “working and con-ceptual methods had spread everywhere”. She could muster high-rankingcolleagues and students who were fascinated by her way of mathematicalthinking.

Nevertheless it seems that there was some opposition against her ab-stract methods, also in Göttingen among the mathematicians. OlgaTaussky-Todd recalls in [45] her impression of the Göttingen mathe-matical scene:

. . . not everybody liked her [Emmy Noether], and not everybody trusted thather achievements were what they later were accepted to be.

One day Olga Taussky had been present when one of the senior professorstalked very roughly to Emmy Noether. (Later he apologized to her forthis insult.) When Emmy Noether had her 50th birthday in 1932 then,as Olga Taussky recalls, nobody at Göttingen had taken notice of it,although at that time all birthdays were published in the Jahresberichtof the DMV.36 Reading all this, I can understand Emmy Noether whenlater in 1935 she said to Veblen about her time in the USA:

The last year and a half had been the very happiest in her whole life, for she wasappreciated in Bryn Mawr and Princeton as she had never been appreciated inher own country.37

Thus it seems that Weyl’s statement that “her scientific merits andscientific greatness by that time was denied by no one” did not describethe situation exactly. Perhaps, since Weyl was the “premier professor”of mathematics at Göttingen, and since he was known to respect and

36 But Hasse in Marburg had sent her a birthday cake, together with a paper whichhe had dedicated to her. The paper was [12]. See [22].

37 Cited from a letter of Abraham Flexner to President Park of Bryn Mawr; see [37].

306 Peter Roquette

acknowledge Noether’s merits and scientific greatness, nobody dared totell him if he disagreed. Olga Taussky-Todd remembers that

“outside of Göttingen, Emmy was greatly appreciated in her country.”

We may add that this was not only so in her country but also world-wide.And of course also in Göttingen there was an ever-growing fraction ofmathematicians, including Weyl, who held Noether in high esteem.

As to Hermann Weyl, let us cite MacLane who was a student at Göt-tingen in the period 1931-33. We read in [24]:

When I first came to Göttingen I spoke to Professor Weyl and expressed my in-terest in logic and algebra. He immediately remarked that in algebra Göttingenwas excellently represented by Professor Noether; he recommended that I at-tend her courses and seminars . . . By the time of my arrival she was Ausseror-dentlicher Professor. However, it was clear that in the view of Weyl, Hilbert,and the others, she was right on the level of any of the full professors. Herwork was much admired and her influence was widespread.

MacLane sometimes joined the hiking parties (Ausflug) of EmmyNoether and her class to the hills around Göttingen. Noether usedthese hiking parties to discuss “algebra, other mathematical topics andRussia”.38 It seems that Weyl too joined those excursions occasionally.There is a nice photo of Noether with Weyl and family, together witha group of mathematicians posing in front of the “Gasthof Vollbrecht”.The photo is published in [5] and dated 1932. Since Artin is seen as amember of the hiking party, it seems very probable that the photo wastaken on the occasion of Artin’s famous Göttingen lectures on class fieldtheory which took place from February 29 to March 2, 1932 .39 Thiswas a big affair and a number of people came from various places inorder to listen to Artin lecturing on the new face of class field theory.The lectures were organized by Emmy Noether. Since she was not afull professor and, accordingly, had no personal funds to organize suchmeetings we suppose that one of her colleagues, probably Weyl, hadmade available the necessary financial means for this occasion. In anycase we see that by now she was able to get support for her activities inGöttingen, not only for the Artin lectures but also for other speakers.

The International Congress of Mathematicians took place in Septem-ber 1932 in Zürich. Emmy Noether was invited to deliver one of the

38 1928/29 Emmy Noether had been in Moscow as a visiting professor, on the invi-tation of Alexandroff whom she knew from Göttingen.

39 The photo is also contained in the Oberwolfach photo collection online. Probablythe photo was taken by Natascha Artin, the wife of Emil Artin.


main lectures there. Usually, proposals for invited speakers at the IMUconferences were submitted by the presidents of the national mathemat-ical organizations which were members of the IMU. In 1931/32 Her-mann Weyl was president (“Vorsitzender”) of the DMV. So it appearsthat Weyl had his hand in the affair when it came to proposing EmmyNoether as a speaker from Germany. The proposal had to be acceptedby the executive committee. The nomination of Emmy Noether wasaccepted and this shows the great respect and admiration which EmmyNoether enjoyed on the international scale.

Emmy Noether’s Zürich lecture can be considered as the high pointin her mathematical career.

6 Göttingen exodus: 1933

The year 1933 brought about the almost complete destruction of theunique mathematical scene in Göttingen. In consequence of the anti-semitic political line of the Nazi government many scientists of Jewishorigin had to leave the university, as well as those who were known to becritical towards the new government. The Göttingen situation in 1933has often been described, and so we can refer to the literature, e.g., [40],[42].

Emmy Noether was of Jewish origin and so she too was a victim ofthe new government policy. On May 5, 1933 Emmy Noether obtainedthe message that she was put “temporarily on leave” from lecturing atthe university. When Hasse heard this, he wrote a letter to her; we donot know the text of his letter but from her reply we may conclude thathe asked whether he could be of help. Emmy Noether replied on May10, 1933:

Lieber Herr Hasse!Vielen herzlichen Dank für Ihren guten freundschaftlichen Brief! Die Sache ist aberdoch für mich sehr viel weniger schlimm als für sehr viele andere: rein äußerlichhabe ich ein kleines Vermögen (ich hatte ja nie Pensionsberechtigung), sodaß icherst einmal in Ruhe abwarten kann; im Augenblick, bis zur definitiven Entschei-dung oder etwas länger, geht auch das Gehalt noch weiter. Dann wird wohljetzt auch einiges von der Fakultät versucht, die Beurlaubung nicht definitiv zumachen; der Erfolg ist natürlich im Moment recht fraglich. Schließlich sagte Weylmir, daß er schon vor ein paar Wochen, wo alles noch schwebte, nach Princetongeschrieben habe wo er immer noch Beziehungen hat. Die haben zwar wegen derDollarkrise jetzt auch keine Entschlußkraft; aber Weyl meinte doch daß mit derZeit sich etwas ergeben könne, zumal Veblen im vorigen Jahr viel daran lag, michmit Flexner, dem Organisator des neuen Instituts, bekannt zu machen. Vielleicht

308 Peter Roquette

kommt einmal eine sich eventuell wiederholende Gastvorlesung heraus, und imübrigen wieder Deutschland, das wäre mir natürlich das liebste. Und vielleichtkann ich Ihnen sogar auch einmal so ein Jahr Flexner-Institut verschaffen - das istzwar Zukunftsphantasie - wir sprachen doch im Winter davon . . .

Dear Mr. Hasse!Thanks very much for your good, friendly letter! But for myself, the situationis much less dire than for many others: in fact I have a small fortune (afterall I was never entitled to pension) and hence for the time being I can quietlywait and see. Also, the salary payments continue until the final decision oreven somewhat longer. Moreover the Faculty tries to avert my suspension tobecome final; at the moment, however, there is little hope for success. Finally,Weyl told me that some weeks ago already when things were still open, he hadwritten to Princeton where he still has contacts. At the moment, however,because of the dollar crisis they don’t have much freedom there for their deci-sions; but Weyl believes that in the course of time there may arise something,in particular since Veblen last year was eager to introduce me to Flexner, theorganizer of the new Institute. Perhaps there will emerge a visiting professor-ship which may be iterated, and in the meantime Germany again, this wouldbe the best solution for me, naturally. And maybe I will be able to manage foryou too a year in the Flexner Institute – but this is my fantasy for the future– we have talked last winter about this. . .

The first impression while reading this letter is her complete selfless-ness, which is well-known from other reports on her life and which ismanifest here again. She does not complain about her own situationbut only points out that for other people things may be worse. Readingfurther, we see that the Faculty in Göttingen tries to keep her; this showsthat she was respected there as a scientist and teacher although she stilldid not have a tenured position. Hermann Weyl was a full professor andhence a member of the Faculty committee; we can surely assume thathe was one of the driving forces in trying to save Emmy Noether for aposition in Göttingen. In fact, in his memorial speech [53] Weyl said:

It was attempted, of course, to influence the Ministerium and other responsibleand irresponsible bodies so that her position might be saved. I suppose therecould hardly have been in any other case such a pile of enthusiastic testimonialsfiled with the Ministerium as was sent in on her behalf. At that time we reallyfought; there was still hope left that the worst could be warded off . . .

And finally, in the above Noether letter we read that, independent ofthese attempts, Weyl had written to Princeton on her behalf. We do notknow whom in Princeton Weyl had adressed. Since Noether mentions inher letter Veblen and Flexner, it seems probable that Weyl had writtento one or both of them. Abraham Flexner was the spiritual founder andthe first director of the newly-founded Institute for Advanced Study in


Princeton. Oswald Veblen was the first permanent mathematics pro-fessor of the IAS. Certain indications suggest that Weyl had written toLefschetz too; see next section. Solomon Lefschetz had the position offull professor at Princeton University.

One year earlier, in the late summer of 1932, Weyl had rejected anoffer to join the IAS as a permanent member. But now, since the po-litical situation had deteriorated, he inquired whether it was possible toreverse his decision. (It was.) From Noether’s letter we infer that Weyldid not only write on his own behalf but also on Noether’s. This factalone demonstrates the very high esteem in which he held Noether as amathematician and as a personality.40

But of course, the best solution would be that Noether could stay inGöttingen. This was what Weyl wished to achieve foremost, as we citedabove. (It was in vain.) Weyl reports in [53]:

I have a particularly vivid recollection of these months. Emmy Noether, hercourage, her frankness, her unconcern about her own fate, her conciliatoryspirit, were, in the middle of all the hatred and meanness, despair and sorrowsurrounding us, a moral solace.

That stormy time of struggle in the summer of 1933 in Göttingen drewthem closer together. This is also evident from the words Weyl used twoyears later in his speech at her funeral:41

You did not believe in evil, indeed it never occurred to you that it could playa role in the affairs of man. This was never brought home to me more clearlythan in the last summer we spent together in Göttingen, the stormy summer of1933. In the midst of the terrible struggle, destruction and upheaval that wasgoing on around us in all factions, in a sea of hate and violence, of fear anddesperation and dejection - you went your own way, pondering the challengesof mathematics with the same industriousness as before. When you were notallowed to use the institute’s lecture halls you gathered your students in yourown home. Even those in their brown shirts were welcome; never for a seconddid you doubt their integrity. Without regard for your own fate, openheartedand without fear, always conciliatory, you went your own way. Many of usbelieved that an enmity had been unleashed in which there could be no pardon;but you remained untouched by it all.

Parallel to the attempts of the Faculty to keep Noether in Göttingen,Hasse took the initiative and collected testimonials42 which would put

40 In the course of time, Weyl used his influence in American academic circles to helpmany other mathematicians as well.

41 See section 9.2.42 The German word is “Gutachten”. I am not sure whether the translation into

“testimonial” is adequate. My dictionary offers also “opinion” or “expertise” or

310 Peter Roquette

into evidence that Emmy Noether was a scientist of first rank and henceit would be advantageous for the scientific environment of Göttingen ifshe did not leave. Hasse collected 14 such testimonials. Together theywere sent to the Kurator of the university who was to forward them tothe Ministerium in Berlin. Recently we have found the text of thosetestimonials which are kept in the Prussian State archives in Berlin; weplan to publish them separately. The names of the authors are:

H. Bohr, KopenhagenPh. Furtwängler, WienG.H. Hardy, CambridgeH. Hasse, MarburgO. Perron, MünchenT. Rella, WienJ.A. Schouten, DelftB. Segre, BolognaK. Shoda, OsakaC. Siegel, FrankfurtA. Speiser, ZürichT. Takagi, TokyoB.L. van der Waerden, LeipzigH. Weyl, Göttingen

We see that also Hermann Weyl wrote a testimonial. We have included itin the appendix, translated into English; see section 9.1. Note that Weylcompared Emmy Noether to Lise Meitner, the nuclear physicist. In thepresent situation this comparison may have been done since Meitner, alsoof Jewish origin, was allowed to stay in Berlin continuing her researchwith Otto Hahn in their common laboratory. After all, the initiatives ofHasse and of Weyl were to obtain a similar status for Emmy Noether inGöttingen.

As is well-known, this was in vain. Perhaps those testimonials werenever read after the Kurator of Göttingen University wrote to the Min-isterium that Emmy Noether’s political opinions were based on “Marx-ism”.43

Let us close this section with some lines from a letter of Weyl toHeinrich Brandt in Halle. The letter is dated December 15, 1933; at that

“letter of recommendation”. I have chosen “testimonial” since Weyl uses thisterminology.

43 See [47]. – By the way, there was another such initiative started, namely in favorof Courant who also had been “beurlaubt” from Göttingen University. That wassigned by 28 scientists including Hermann Weyl and Helmut Hasse. Again this wasnot successful, although this time the Kurator’s statement was not as negative asin Noether’s case. (We have got this information from Constance Reid’s book onCourant [34].)


time Weyl and Noether were already in the USA. Brandt was known tobe quite sceptical towards abstract methods in mathematics; he did noteven like Artin’s beautiful presentation of his own (Brandt’s) discovery,namely that the ideals and ideal classes of maximal orders in a simplealgebra over a number field form a groupoid under multiplication.44

(The notion of “groupoid” is Brandt’s invention.) Weyl’s letter is a replyto one from Brandt which, however, is not known to us. ApparentlyBrandt had uttered some words against Noether’s abstracting method,and Weyl replied explaining his own viewpoint:45

. . . So wenig mir persönlich die “abstrakte” Algebra liegt, so schätze ich doch ihreLeistungen und ihre Bedeutung offenbar wesentlich höher ein, als Sie das tun. Esimponiert mir gerade an Emmy Noether, daß ihre Probleme immer konkreter undtiefer geworden sind.

Personally, the “abstract” algebra doesn’t suit me well, but apparently I doestimate its achievements and importance much higher than you are doing. Iam particularly impressed that Emmy Noether’s problems have become moreand more concrete and deep.

Weyl continues as follows. It is not known whether Brandt had writtensome comments on Noether’s Jewish origin and connected this with herabstract way of thinking, or perhaps Weyl’s letter was triggered by thegeneral situation in Germany and especially in Göttingen:

Warum soll ihr, der Hebräerin, nicht zustehen, was in den Händen des “Ariers”Dedekind zu großen Ergebnissen geführt hat? Ich überlasse es gern Herrn Spen-gler und Bieberbach, die mathematische Denkweise nach Völkern und Rassenzu zerteilen. Daß Göttingen den Anspruch verloren hat, mathematischer Vorortzu sein, gebe ich Ihnen gerne zu – was ist denn überhaupt von Göttingen übriggeblieben? Ich hoffe und wünsche, daß es eine seiner alten Tradition würdigeFortsetzung durch neue Männer finden möge; aber ich bin froh, daß ich es nichtmehr gegen einen Strom von Unsinn und Fanatismus zu stützen brauche!

Why should she, as of Hebrew descent, not be entitled to do what had ledto such great results in the hands of Dedekind, the “Arian” ? I leave it toMr. Spengler and Mr. Bieberbach to divide the mathematical way of thinkingaccording to nations and races. I concede that Göttingen has lost its role as ahigh-ranking mathematical place – what is actually left of Göttingen? I hopeand wish that Göttingen would find a continuation by new men, worthy of itslong tradition; but I am glad that I do not have to support it against a torrentof nonsense and fanaticism.

44 Artin’s paper is [3].45 I would like to thank M. Göbel for sending me copies of this letter from the Brandt

archive in Halle. The letter is published in [17], together with other letters Brandt-Weyl and Brandt-Noether.

312 Peter Roquette

7 Bryn Mawr: 1933-1935

As we have seen in the foregoing section, Weyl had written to Princetonon behalf of Emmy Noether, and this was in March or April 1933 already.Since he was going to join the Institute for Advanced Study in Princeton,one would assume that he had recommended accepting Emmy Noetheras a visiting scientist of the Institute. We know that some people atthe Institute were interested in getting Noether to Princeton, for atthe International Zürich Congress Oswald Veblen had been eager tointroduce Emmy Noether to the Institute’s director, Abraham Flexner.(See Noether’s letter to Hasse, cited in the foregoing section.)

But as it turned out, Emmy Noether did not receive an invitation as avisitor to the Institute. We do not know the reason for this; perhaps theimpending dollar crisis, mentioned in Noether’s letter to Hasse, forcedthe Institute to reduce its available funds. Or, may there have beenother reasons as well? On the other hand, from the documents whichwe found in the archive of Bryn Mawr College it can be seen that theInstitute for Advanced Study contributed a substantial amount towardsthe salary of Emmy Noether in Bryn Mawr.

We do not know who was the first to suggest that Bryn Mawr Collegecould be a suitable place for Emmy Noether. Some evidence points to theconclusion that it was Solomon Lefschetz. In fact, we have found a letter,dated June 12, 1933 already, adressed to the “Emergency Committee inAid of Displaced German Scholars”, where he discusses future aspectsfor Emmy Noether and proposes Bryn Mawr.46 Lefschetz had visitedGöttingen two years ago and so he knew Emmy Noether personally.Lefschetz’ letter is quite remarkable since, firstly, he clearly expressesthat Emmy Noether, in his opinion, was a leading figure in contemporarymathematics; secondly we see that he had taken already practical stepsto provide Bryn Mawr with at least part of the necessary financial meansin order to offer Emmy Noether a stipend. Let us cite the relevantportions of that letter:

Dear Dr. Duggan: I am endeavoring to make connections with some wealthypeople in Pittsburgh, one of them a former Bryn Mawr student, with a viewof raising a fund to provide a research associateship at Bryn Mawr for MissEmmy Noether. As you may know, she is one of the most distinguished victimsof the Hitler cold pogrom and she is victimized doubly; first for racial reasonsand second, owing to her sex. It occured to me that it would be a fine thingto have her attached to Bryn Mawr in a position which would compete with

46 We have found this letter in the archives of the New York Public Library.


no one and would be created ad hoc; the most distinguished feminine math-ematician connected with the most distinguished feminine university. I havecommunicated with Mrs. Wheeler, the Head of the Department at Bryn Mawr,and she is not only sympathetic but thoroughly enthusiastic for this plan,

So far as I know, your organization is the only one which is endeavoringto do anything systematic to relieve the situation of the stranded German sci-entists. As I do not think random efforts are advisable, I wish first of all toinform you of my plan. Moreover, if I were to succeed only partially, would itbe possible to get any aid from your organization? I would greatly appreciateyour informing me on this point at your earliest convenience.

In the preliminary communication with my intended victims I mentionedthe following proposal: to contribute enough annually to provide Miss Noetherwith a very modest salary, say $ 2000, and a retiring allowance of $ 1200.

Yours very sincerely, S. Lefschetz.Already one month later the committee granted the sum of $ 2000 toBryn Mawr for Emmy Noether.

There arises the question from whom Lefschetz had got the infor-mation, at that early moment already, that Emmy Noether had beensuspended.47 We are inclined to believe that it was Hermann Weyl. Ido not know whether the correspondence of Lefschetz of those years hasbeen preserved in some archive, and where. Perhaps it will be possibleto find those letters and check.

Emmy Noether arrived in Bryn Mawr in early November 1933. Herfirst letter from Bryn Mawr to Hasse is dated March 6, 1934. She re-ported, among other things, that since February she gave a lecture oncea week at the Institute for Advanced Study in Princeton. In this lectureshe had started with representation modules and groups with operators.She mentions that Weyl too is lecturing on representation theory, andthat he will switch to continuous groups later. It appears that the Göt-tingen situation of 1926/27 was repeating. And we imagine HermannWeyl and Emmy Noether walking after her lectures around the Campusof Princeton University48 instead of Göttingen’s narrow streets, vividlydiscussing new aspects of representation theory.

In the book [34] on Courant we read:Weyl sent happy letters from Princeton. In Fine Hall, where Flexner’s groupwas temporarily housed, German was spoken as much as English. He frequentlysaw Emmy Noether . . .

47 Emmy Noether had been “beurlaubt”, i.e., temporarily suspended from her du-ties, in May 1933. Observing that mail from Europe to USA used about 2-3weeks at that time, we conclude that Lefschetz must have started working on hisNoether-Bryn Mawr idea immediately after receiving the news about her suspen-sion. Noether was finally dismissed from university on September 9, 1933.

48 The Institute’s Fuld Hall had not yet been built and the School of Mathematicsof the Institute was temporarily housed in Fine Hall on the University Campus.

314 Peter Roquette

Perhaps in the Courant legacy we can find more about Weyl and Noetherin Princeton, but we have not been able yet to check those sources.

Every week Emmy Noether visited the Brauers in Princeton; RichardBrauer was assistant to Weyl in that year and perhaps sometimes Her-mann Weyl also joined their company. The name of Hermann Weylappears several times in her letters to Hasse from Bryn Mawr. In Novem-ber 1934 she reports that she had studied Weyl’s recent publication onRiemann matrices in the Annals of Mathematics.

Emmy Noether died on April 14, 1935. One day later Hermann Weylcabled to Hasse:

hasse mathematical institute gottingen – emmy noether died yesterday – bysudden collapse after successful – operation of tumor 49 few days ago – burialwednesday bryn mawr – weyl

At the burial ceremony on Wednesday Weyl spoke on behalf of her Ger-man friends and colleagues. We have included an English translationof this moving text in the appendix; see section 9.2. One week laterhe delivered his memorial lecture in the large auditorium of Bryn MawrCollege. That text is published and well known [53].

8 The Weyl-Einstein letter to the NYT

On Sunday May 5, 1935 the New York Times published a “Letter to theEditor”, signed by Albert Einstein and headed by the following title:

Professor Einstein Writes in Appreciation of a Fellow-Mathematician.

We have included the text of this letter in our appendix; see section 9.3.Reading this letter one is struck by the almost poetic style which el-

evates the text to one of the pearls in the literature on mathematics.The text is often cited, the last citation which I found is in the “Mit-teilungen” of the DMV, 2007 where Jochen Brüning tries to connectmathematics with poetry [6]. But because of this character of style ithas been doubted whether the text really was composed by Einsteinhimself. If not then this would not have been the first and not thelast incident where Einstein had put his name under a text which wasnot conceived by himself – provided that in his opinion the subject was

49 President Park of Bryn Mawr had sent a detailed report, dated May 16, 1935, toOtto Nöther in Mannheim, a cousin of Emmy Noether. A copy of that letter ispreserved. There it is stated that according to the the medical diagnosis of thedoctors who operated her, Emmy Noether suffered from a “pelvis tumor”.


worth-while to support. Since Weyl’s poetic style was known it was notconsidered impossible that the text was composed by Hermann Weyl.

Some time ago I have come across a letter signed by Dr. Ruth Stauffer-McKee. I include a copy of that letter in the appendix; see section9.4. In particular I refer to the last paragraph of the letter. Basedon the information provided by Stauffer I came to the conclusion that,indeed, the text was essentially written by Weyl. I have expressed thisopinion in my talk in Bielefeld and also in a “Letter to the Editor” ofthe “Mitteilungen der Deutschen Mathematiker-Vereinigung” [38].

However, recently I have been informed that Einstein’s draft of thisletter in his own handwriting has been found by Siegmund-Schultze50 inthe Einstein archive in Jerusalem. The article is to appear in the nextissue of the Mitteilungen der DMV [43]. This then settles the questionof authorship in favor of Einstein. But what had induced Ruth Staufferto claim that Weyl had “inspired” Einstein’s letter?

In order to understand Stauffer’s letter let us explain its background.In 1972 there appeared a paper on Emmy Noether in the Ameri-

can Mathematical Monthly, authored by Clark Kimberling [18]. Amongother information the paper contains the text of Einstein’s letter to theNew York Times. Kimberling had obtained the text from an article inthe Bryn Mawr Alumnae Bulletin where it had been reprinted in 1935.Together with that text, we find in [18] the following:

A note in the files of the Bryn Mawr Alumnae Bulletin reads, “The above wasinspired, if not written, by Dr. Hermann Weyl, eminent German mathemati-cian. Mr. Einstein had never met Miss Noether.”

(Here, by “above” was meant the text of the Einstein letter to the NewYork Times.)

While the first sentence of that “note” can be considered as an affirma-tion of the guess that Weyl had conceived the text of Einstein’s letter,the second sentence is hard to believe. Emmy Noether often visited theInstitute for Advanced Study in Princeton, the same place where Ein-stein was, and it seems improbable that they did not meet there. Afterall, Einstein was already in May 1918 well informed about Noether’sachievements, when he wrote to Hilbert praising her work [26]. And inDecember that year, after receiving the printed version of this work, hewrote to Felix Klein and recommended her Habilitation. In the 1920s,Einstein had a correspondence with Emmy Noether who acted as referee

50 I would like to thank R. Siegmund-Schultze for a number of interesting commentsand corrections to this article.

316 Peter Roquette

for papers which were submitted to the Mathematische Annalen. It ishard to believe that in Princeton he would have avoided meeting EmmyNoether, whom he esteemed so highly. Moreover, we have already men-tioned in section 4 that Einstein probably had met Noether in 1915 inGöttingen. Also, on the DMV-meetings 1909 in Salzburg and 1913 inWien both Einstein and Emmy Noether presented talks and there wasample opportunity for them to meet.

Thus it seemed that the “note” which Kimberling mentioned had beenwritten by someone who was not well informed about the situation inthe early thirties. Actually, that “note” was not printed in the BrynMawr Alumnae Bulletin but it was added later by typewriter, maybeonly on the copy which was sent to Kimberling. It is not known whohad been the author of that “note”.

In the same volume of the American Mathematical Monthly wherehis article [18] had appeared, Kimberling published an Addendum say-ing that Einstein’s former secretary, Miss Dukas, had objected to thestatement that the letter written by Einstein was “inspired, if not writ-ten by Dr. Hermann Weyl”. She insisted that the letter was written byEinstein himself at the request of Weyl.

This, however, induced Ruth Stauffer to write the above mentionedletter to the editor of the American Mathematical Monthly, which weare citing in section 9.4. Ruth Stauffer had been a Ph.D. student ofEmmy Noether in Bryn Mawr and in her letter she recalls vividly themathematical atmosphere in Princeton at that time.

On this evidence we were led to believe that the statement of Ein-stein’s secretary Dukas may be due to a mix-up on her part. For, onlyshortly before Noether’s death Einstein had written another letter inwhich he recommends that Emmy Noether’s situation in Bryn MawrCollege should be improved and put on a more solid base. At that timePresident Park of Bryn Mawr had tried to obtain testimonies on EmmyNoether, which could be used in order to get funds for a more perma-nent position.51 Einstein’s testimony is dated January 8, 1935 and iswritten in German; we have found it in the archives of the Institute forAdvanced Study in Princeton. Its full text reads:

Fräulein Dr. Emmy Noether besitzt unzweifelhaft erhebliches schöpferisches Tal-ent, was jeweilen von nicht sehr vielen Mathematikern einer Generation gesagt wer-den kann. Ihr die Fortsetzung der wissenschaftlichen Arbeit zu ermöglichen, be-

51 This was successful, but Emmy Noether died before she got to know about it. –Other testimonials, by Solomon Lefschetz, Norbert Wiener and George D. Birkhoffare published in Kimberling’s article [19].


deutet nach meiner Ansicht die Erfüllung einer Ehrenpflicht und wirkliche Förderungwissenschaftlicher Forschung.

Without doubt Miss Dr. Emmy Noether commands significant and creativetalent; this cannot be said of many mathematicians of one generation. In myopinion it is an obligation of honor to provide her with the means to con-tinue her scientific work, and indeed this will be a proper support of scientificresearch.

It is apparent that the style of this is quite different from the style ofthe letter to the New York Times.

Although we now know that Miss Dukas was right and Einstein hadcomposed his NYT-letter with his own hand, there remains the questionas to the basis of Stauffer’s contentions.

Stauffer was a young student and what she reports is partly basedon what she heard from Mrs. Wheeler. But the latter, who was headof the mathematics department of Bryn Mawr College at the time, hadstudied in Göttingen with Hilbert in the same years as Hermann Weylhad; so they were old acquaintances and it seems probable that Weylhimself had told her the story as it had happened. Thus it may wellhave been that first Weyl had sent his obituary on Emmy Noether tothe New York Times, and that this was returned with the suggestionthat Einstein should write an obituary – as Ruth Stauffer narrates. Andthen Einstein wrote his letter “at the request of Weyl”, as Miss Dukas hasclaimed. Whether there was any cooperation between Einstein and Weylwhile drafting the letter is not known. But we can safely assume thatboth had talked if not about the text of the letter but certainly aboutEmmy Noether’s personality, her work and her influence on mathematicsat large. In this way Stauffer’s claim may be justified that Weyl had“inspired” Einstein in writing his letter.

Remark: It has been pointed out to me by several people that thevery last sentence in the English version of Einstein’s letter deviates in itsmeaning from the original German text wheras otherwise the translationseems to be excellent.52 In the English version it is said that Noether’slast years in Bryn Mawr were made the “happiest and perhaps mostfruitful years of her entire career”, but the German text does not referto her entire career and only pointed out that death came to her “mittenin froher und fruchtbarer Arbeit”. I do not know who had translated theGerman text into English. There is a letter of Abraham Flexner, thedirector of the Institute for Advanced Study in Princeton, addressed toEinstein and dated April 30, 1935, in which Flexner thanks Einstein for

52 The German text is published in my “Letter to the Editor” [38].

318 Peter Roquette

the “beautiful tribute to Miss Noether” and continues: “I shall translateit into English and send it to the New York Times, through which it willreach, I think, many of those who should know of her career.” But it doesnot seem justified, I believe, to conclude that Flexner personally did thetranslation job. He was quite busy with all kinds of responsibilities andcertainly he had contacts to experts who would have been willing andcompetent to do it.53

Final Remark: Weyl’s solidarity with Emmy Noether extended toher brother and family. Emmy’s brother Fritz had emigrated to Russiawhere he got a position at the university in Tomsk. In 1937 he wasarrested and sentenced to 25 years in prison because of alleged espionagefor Germany. In the Einstein archive in Jerusalem we have found aletter, dated April 1938 and signed by Einstein, addressed to the Russianminister of foreign affairs Litvinov. In this letter Einstein appeals to theminister in favor of Fritz Noether, whom he (Einstein) is sure to beinnocent. In the Einstein archive, right after this letter, is preserved acurriculum vitae of Fritz Noether in Weyl’s handwriting. Thus again itappears that Weyl has “inspired” Einstein to write such a letter.54

Among Weyl’s papers I found a number of letters from 1938 and thefollowing years, which show that he cared for the two sons of FritzNoether, Hermann and Gottfried, who had to leave the Soviet Unionafter their father had been sentenced. Weyl saw to it that they obtainedimmigrant visa to the United States, and that they got sufficient meansto finance their university education. Both became respected membersof the scientific community.

9 Appendix: documents

9.1 Weyl’s testimony

The following text 55 is from the testimonial, signed by Hermann Weyl onJuly 12, 1933 and sent by Hasse to the Ministerium in Berlin togetherwith 13 other testimonials. We have found these testimonials in thePrussian state archive Berlin.

53 Siegmund-Schultze [43] advocates reasons to assume that indeed, Flexner himselfdid the translation job.

54 The appeal of Einstein was in vain. In 1941, when German troops were approach-ing the town of Orjol where Fritz was kept in prison, he was sentenced to deathand immediately executed. See, e.g., [41].

55 Translated from German by Ian Beaumont.


Emmy Noether has attained a prominent position in current mathe-matical research – by virtue of her unusual deep-rooted prolific power,and of the central importance of the problems she is working on togetherwith their interrelationships. Her research and the promising nature ofthe material she teaches enabled her in Göttingen to attract the largestgroup of students. When I compare her with the two woman mathe-maticians whose names have gone down in history, Sophie Germain andSonja Kowalewska, she towers over them due to the originality and in-tensity of her scientific achievements. The name Emmy Noether is asimportant and respected in the field of mathematics as Lise Meitner isin physics.

She represents above all “Abstract Algebra”. The word “abstract” inthis context in no way implies that this branch of mathematics is ofno practical use. The prevailing tendency is to solve problems usingsuitable visualizations, i.e. appropriate formation of concepts, ratherthan blind calculations. Fräulein Noether is in this respect the legitimatesuccessor of the great German number theorist R. Dedekind. In addition,Quantum Theory has made Abstract Algebra the area of mathematicsmost closely related to physics.

In this field, in which mathematics is currently experiencing its mostactive progress, Emmy Noether is the recognised leader, both nationallyand internationally.

Hermann Weyl

9.2 Weyl’s funeral speech

The following text56 was spoken by Hermann Weyl on Emmy Noether’sfuneral on April 18, 1935. We have found this text in the legacy of GreteHermann, which is preserved in the “Archiv der sozialen Demokratie” inBonn.

The hour has come, Emmy Noether, in which we must forever takeour leave of you. Many will be deeply moved by your passing, none moreso than your beloved brother Fritz, who, separated from you by half theglobe, was unable to be here, and who must speak his last farewell toyou through my mouth. His are the flowers I lay on your coffin. We bowour heads in acknowledgement of his pain, which it is not ours to putinto words.

But I consider it a duty at this hour to articulate the feelings of your

56 translated from German by Ian Beaumont

320 Peter Roquette

German colleagues - those who are here, and those in your homelandwho have held true to our goals and to you as a person. I find it apt,too, that our native tongue be heard at your graveside - the languageof your innermost sentiments and in which you thought your thoughts -and which we hold dear whatever power may reign on German soil. Yourfinal rest will be in foreign soil, in the soil of this great hospitable countrythat offered you a place to carry on your work after your own countryclosed its doors on you. We feel the urge at this time to thank Americafor what it has done in the last two years of hardship for German science,and to thank especially Bryn Mawr, where they were both happy andproud to include you amongst their teachers.

Justifiably proud, for you were a great woman mathematician - I haveno reservations in calling you the greatest that history has known. Yourwork has changed the way we look at algebra, and with your manygothic letters you have left your name written indelibly across its pages.No-one, perhaps, contributed as much as you towards remoulding theaxiomatic approach into a powerful research instrument, instead of amere aid in the logical elucidation of the foundations of mathematics,as it had previously been. Amongst your predecessors in algebra andnumber theory it was probably Dedekind who came closest.

When, at this hour, I think of what made you what you were, twothings immediately come to mind . The first is the original, productiveforce of your mathematical thinking. Like a too ripe fruit, it seemed toburst through the shell of your humanness. You were at once instrumentof and receptacle for the intellectual force that surged forth from withinyou. You were not of clay, harmoniously shaped by God’s artistic hand,but a piece of primordial human rock into which he breathed creativegenius.

The force of your genius seemed to transcend the bounds of yoursex - and in Göttingen we jokingly, but reverentially, spoke of you inthe masculine, as "den Noether". But you were a woman, maternal,and with a childlike warmheartedness. Not only did you give to yourstudents intellectually - fully and without reserve - they gathered roundyou like chicks under the wings of a mother hen; you loved them, caredfor them and lived with them in close community.

The second thing that springs to mind is that your heart knew nomalice; you did not believe in evil, indeed it never occurred to you thatit could play a role in the affairs of man. This was never brought hometo me more clearly than in the last summer we spent together in Göttin-gen, the stormy summer of 1933. In the midst of the terrible struggle,


destruction and upheaval that was going on around us in all factions, ina sea of hate and violence, of fear and desperation and dejection - youwent your own way, pondering the challenges of mathematics with thesame industriousness as before. When you were not allowed to use theinstitute’s lecture halls you gathered your students in your own home.Even those in their brown shirts were welcome; never for a second didyou doubt their integrity. Without regard for your own fate, openheartedand without fear, always conciliatory, you went your own way. Many ofus believed that an enmity had been unleashed in which there could beno pardon; but you remained untouched by it all. You were happy to goback to Göttingen last summer, where, as if nothing had happened, youlived and worked with German mathematicians striving for the samegoals. You planned on doing the same this summer.

You truly deserve the wreath that the mathematicians in Göttingenhave asked me to lay on your grave.

We do not know what death is. But is it not comforting to thinkthat souls will meet again after this life on Earth, and how your father’ssoul will greet you? Has any father found in his daughter a worthiersuccessor, great in her own right?

You were torn from us in your creative prime; your sudden departure,like the echo of a thunderclap, is still written on our faces. But yourwork and your disposition will long keep your memory alive, in scienceand amongst your students, friends and colleagues.

Farewell then, Emmy Noether, great mathematician and great woman.Though decay take your mortal remains, we will always cherish thelegacy you left us.

Hermann Weyl

9.3 Letter to the New York Times

The following text was published on Sunday, May 5, 1935 by the NewYork Times, with the heading: “Professor Einstein Writes in Apprecia-tion of a Fellow-Mathematician”.

To the Editor of The New York Times:The efforts of most human-beings are consumed in the struggle for

their daily bread, but most of those who are, either through fortuneor some special gift, relieved of this struggle are largely absorbed infurther improving their worldly lot. Beneath the effort directed towardthe accumulation of worldly goods lies all too frequently the illusion

322 Peter Roquette

that this is the most substantial and desirable end to be achieved; butthere is, fortunately, a minority composed of those who recognize earlyin their lives that the most beautiful and satisfying experiences open tohumankind are not derived from the outside, but are bound up with thedevelopment of the individual’s own feeling, thinking and acting. Thegenuine artists, investigators and thinkers have always been persons ofthis kind. However inconspicuously the life of these individuals runs itscourse, none the less the fruits of their endeavors are the most valuablecontributions which one generation can make to its successors.

Within the past few days a distinguished mathematician, ProfessorEmmy Noether, formerly connected with the University of Göttingenand for the past two years at Bryn Mawr College, died in her fifty-thirdyear. In the judgment of the most competent living mathematicians,Fräulein Noether was the most significant creative mathematical geniusthus far produced since the higher education of women began. In therealm of algebra, in which the most gifted mathematicians have beenbusy for centuries, she discovered methods which have proved of enor-mous importance in the development of the present-day younger gener-ation of mathematicians. Pure mathematics is, in its way, the poetry oflogical ideas. One seeks the most general ideas of operation which willbring together in simple, logical and unified form the largest possiblecircle of formal relationships. In this effort toward logical beauty spir-itual formulas are discovered necessary for the deeper penetration intothe laws of nature.

Born in a Jewish family distinguished for the love of learning, EmmyNoether, who, in spite of the efforts of the great Göttingen mathemati-cian, Hilbert, never reached the academic standing due her in her owncountry, none the less surrounded herself with a group of students andinvestigators at Göttingen, who have already become distinguished asteachers and investigators. Her unselfish, significant work over a periodof many years was rewarded by the new rulers of Germany with a dis-missal, which cost her the means of maintaining her simple life and theopportunity to carry on her mathematical studies. Farsighted friends ofscience in this country were fortunately able to make such arrangementsat Bryn Mawr College and at Princeton that she found in America upto the day of her death not only colleagues who esteemed her friendshipbut grateful pupils whose enthusiasm made her last years the happiestand perhaps the most fruitful of her entire career.

Albert Einstein.Princeton University, May 1, 1935.


9.4 Letter of Dr. Stauffer-McKee

The following letter was sent by Dr. Ruth Stauffer-McKee on October 17,1972 to the editor of the American Mathematical Monthly, Professor H.Flanders. A carbon copy had been sent to Professor Kimberling. I amindebted to Clark Kimberling for giving me access to his private archive.

Dear Mr. Flanders,After reading the Addendum to “Emmy Noether” in the August Septem-

ber issue of the American Mathematical Monthly, I was much disturbedby the apparent lack of information concerning the thirties at Prince-ton! Rechecking the reference to the original article which appeared inFebruary 1972 I was even more disturbed to note that the quote wasattributed to a note in the files of Bryn Mawr Alumnae Bulletin. Atelephone conversation and a careful check by the Staff of the Bulletinassured me that there was nothing in the files of the Bulletin to evenimply that “Mr. Einstein had never met Miss Noether.”

In respect to the “thirties at Princeton”, I should like to note thatthere was an air of continued excitement at the Institute for AdvancedStudy. Solomon Lefschetz, a guiding spirit who worked diligently to helpthe displayed mathematicians, Hermann Weyl, a leading mathematicianof that time who had learned to know Miss Noether in Göttingen, andJohn von Neumann, then considered a brilliant young genius, were all atthe Institute when Einstein arrived in December of 1933. Mrs. Wheeler,of Bryn Mawr, often told of the welcoming party which she and MissNoether attended.

Mrs. Wheeler usually drove Miss Noether to Princeton for lecturesand included Miss Noether’s students in the parties. We listened totalks by these men who were the leaders in new exciting theories. It wasa friendly group and after the talks everyone gathered for more talk andcoffee in a long pleasant common room. There is no doubt that Einsteinand Noether were acquainted. I saw them in the same group!

As regards the quote in the “addendum to ‘Emmy Noether’ " “inspired,if not written by Dr. Hermann Weyl” is certainly true. The writing of theobituary was a very natural occurence. Hermann Weyl was consideredby the mathematicians as the mathematical leader of the time and atthe peak of his productivity and he had probably the greatest knowledgeand understanding of her work. Einstein had begun to slow down andVon Neumann was relatively young and still growing. It was, therefore,obvious to all the mathematicians that Weyl should write the obituary– which he did. He, furthermore, sent it to the New York Times, the

324 Peter Roquette

New York Times asked who is Weyl? Have Einstein write something, heis the mathematician recognized by the world. This is how Einstein’sarticle appeared. It was most certainly “inspired” by Weyl’s draft. Thesefacts were told to me at the time by Mrs. Wheeler who was indignantthat the New York Times had not recognized the mathematical statureof Hermann Weyl.

Very truly yours,Ruth Stauffer McKee

Senior Mathematician

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