Harmonic Analysis on Commutative Spaces

http://dx.doi.org/10.1090/surv/142

Mathematical Surveys

and Monographs

Volume 142

Harmonic Analysis on Commutative Spaces

Joseph A. Wolf

Amer ican Mathemat ica l Society

E D I T O R I A L C O M M I T T E E

Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss

J. T. Stafford, Chair

2000 Mathematics Subject Classification. Primary 20G20, 22D10, 22Exx, 53C30, 53C35.

For additional information and updates on this book, visit www.ams.org/bookpages /surv-142

Library of C o n g r e s s Cataloging- in-Publ icat ion D a t a Wolf, Joseph Albert, 1936-

Harmonic analysis on commutative spaces / Joseph A. Wolf. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 142)

Includes bibliographical references and indexes. ISBN 978-0-8218-4289-8 (alk. paper) 1. Harmonic analysis. 2. Topological groups. 3. Abelian groups. 4. Algebraic spaces.

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10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07

To Lois

Contents

Introduction xiii Acknowledgments xv Notational Conventions xv

P a r t 1. GENERAL THEORY OF TOPOLOGICAL GROUPS

Chapter 1. Basic Topological Group Theory 3 1.1. Definition and Separation Properties 3 1.2. Subgroups, Quotient Groups, and Quotient Spaces 4 1.3. Connectedness 5 1.4. Covering Groups 7 1.5. Transformation Groups and Homogeneous Spaces 8 1.6. The Locally Compact Case 9 1.7. Product Groups 12 1.8. Invariant Metrics on Topological Groups 15

Chapter 2. Some Examples 19 2.1. General and Special Linear Groups 19 2.2. Linear Lie Groups 20 2.3. Groups Defined by Bilinear Forms 21 2.4. Groups Defined by Hermitian Forms 22 2.5. Degenerate Forms 25 2.6. Automorphism Groups of Algebras 26 2.7. Spheres, Projective Spaces and Grassmannians 28 2.8. Complexification of Real Groups 30 2.9. p-adic Groups 32 2.10. Heisenberg Groups 33

Chapter 3. Integration and Convolution 35 3.1. Definition and Examples 35 3.2. Existence and Uniqueness of Haar Measure 36 3.3. The Modular Function 41 3.4. Integration on Homogeneous Spaces 44 3.5. Convolution and the Lebesgue Spaces 45

viii C O N T E N T S

3.6. The Group Algebra 48 3.7. The Measure Algebra 50 3.8. Adele Groups 51

P a r t 2. REPRESENTATION THEORY AND COMPACT GROUPS

Chapter 4. Basic Representation Theory 55 4.1. Definitions and Examples 56 4.2. Subrepresentations and Quotient Representations 59 4.3. Operations on Representations 64

4.3A. Dual Space 64 4.3B. Direct Sum 64 4.3C. Tensor Product of Spaces 65 4.3D. Horn 67 4.3E. Bilinear Forms 67 4.3F. Tensor Products of Algebras 68 4.3G. Relation with the Commuting Algebra 69

4.4. Multiplicities and the Commuting Algebra 70 4.5. Completely Continuous Representations 72 4.6. Continuous Direct Sums of Representations 75 4.7. Induced Representations 77 4.8. Vector Bundle Interpretation 81 4.9. Mackey's Little-Group Theorem 82

4.9A. The Normal Subgroup Case 82 4.9B. Cohomology and Projective Representations 84 4.9C. Cocycle Representations and Extensions 85

4.10. Mackey Theory and the Heisenberg Group 87

93 93 96 97 99 101 104 107 107 107 110 111 111 112 113 115

Chapter 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7.

5.8. 5.9.

5. Representations of Compact Groups Finite Dimensionality Orthogonality Relations Characters and Projections The Peter-Weyl Theorem The Plancherel Formula Decomposition into Irreducibles Some Basic Examples 5.7A. The Group 17(1) 5.7B. The Group SU(2) 5.7C. The Group SO(3) 5.7D. The Group 50(4) 5.7E. The Sphere S2

5.7F. The Sphere S3

Real, Complex and Quaternion Representations The Frobenius Reciprocity Theorem

CONTENTS ix

Chapter 6. Compact Lie Groups and Homogeneous Spaces 119 6.1. Some Generalities on Lie Groups 119 6.2. Reductive Lie Groups and Lie Algebras 122 6.3. Cartan's Highest Weight Theory 127 6.4. The Peter-Weyl Theorem and the Plancherel Formula 131 6.5. Complex Flag Manifolds and Holomorphic Vector Bundles 133 6.6. Invariant Function Algebras 136

Chapter 7. Discrete Co-Compact Subgroups 141 7.1. Basic Properties of Discrete Subgroups 141 7.2. Regular Representations on Compact Quotients 146 7.3. The First Trace Formula for Compact Quotients 147 7.4. The Lie Group Case 148

Part 3. INTRODUCTION TO COMMUTATIVE SPACES

Chapter 8. Basic Theory of Commutative Spaces 153 8.1. Preliminaries 153 8.2. Spherical Measures and Spherical Functions 156 8.3. Alternate Formulation in the Differentiable Setting 160 8.4. Positive Definite Functions 165 8.5. Induced Spherical Functions 168 8.6. Example: Spherical Principal Series Representations 170 8.7. Example: Double Transitivity and Homogeneous Trees 174

8.7A. Doubly Transitive Groups 174 8.7B. Homogeneous Trees 175 8.7C. A Special Case 176

Chapter 9. Spherical Transforms and Plancherel Formulae 179 9.1. Commutative Banach Algebras 179 9.2. The Spherical Transform 184 9.3. Bochner's Theorem 187 9.4. The Inverse Spherical Transform 191 9.5. The Plancherel Formula for K\G/K 192 9.6. The Plancherel Formula for G/K 194 9.7. The Multiplicity Free Criterion 197 9.8. Characterizations of Commutative Spaces 198 9.9. The Uncertainty Principle 199

9.9A. Operator Norm Inequalities for K\G/K 199 9.9B. The Uncertainty Principle for K\G/K 202 9.9C. Operator Norm Inequalities for G/K 203 9.9D. The Uncertainty Principle for G/K 204

9.10. The Compact Case 204

x CONTENTS

Chapter 10. Special Case: Commutative Groups 207 10.1. The Character Group 207 10.2. The Fourier Transform and Fourier Inversion Theorems 212 10.3. Pontrjagin Duality 214 10.4. Almost Periodic Functions 216 10.5. Spectral Theorems 218 10.6. The Lie Group Case 219

P a r t 4. STRUCTURE AND ANALYSIS FOR COMMUTATIVE SPACES

Chapter 11. Riemannian Symmetric Spaces 225 11.1. A Fast Tour of Symmetric Space Theory 225

11.1 A. Riemannian Basics 225 11.IB. Lie Theoretic Basics 226 11.1C. Complex and Quaternionic Structures 229

11.2. Classifications of Symmetric Spaces 231 11.3. Euclidean Space 236

11.3A. Construction of Spherical Functions 236 11.3B. General Spherical Functions on Euclidean Space 238 11.3C. Positive Definite Spherical Functions on Euclidean

Space 240 11.3D. The Transitive Case 242

11.4. Symmetric Spaces of Compact Type 245 11.4A. Restricted Root Systems 245 11.4B. The Cartan-Helgason Theorem 246 11.4C. Example: Group Manifolds 249 11.4D. Examples: Spheres and Projective Spaces 250

11.5. Symmetric Spaces of Noncompact Type 252 11.5A. Restricted Root Systems 253 11.5B. Harish-Chandra's Parameterization 254 11.5C. Hyperbolic Spaces 255 11.5D. The c-Function and Plancherel Measure 257 11.5E. Example: Groups with Only One Conjugacy Class of

Cartan Subgroups 258 11.6. Appendix: Finsler Symmetric Spaces 260

Chapter 12. Weakly Symmetric and Reductive Commutative Spaces 263 12.1. Commutativity Criteria 263 12.2. Geometry of Weakly Symmetric Spaces 264 12.3. Example: Circle Bundles over Hermit ian Symmetric Spaces 268 12.4. Structure of Spherical Spaces 272 12.5. Complex Weakly Symmetric Spaces 275 12.6. Spherical Spaces are Weakly Symmetric 277 12.7. Kramer Classification and the Akhiezer-Vinberg Theorem 282 12.8. Semisimple Commutative Spaces 287

CONTENTS xi

12.9. Examples of Passage from the Semisimple Case 290 12.10. Reductive Commutative Spaces 293

Chapter 13. Structure of Commutative Nilmanifolds 299 13.1. The "2-step Nilpotent" Theorem 299

13.1 A. Solvable and Nilpotent Radicals 299 13.1B. Group Theory Proof 300 13.1C. Digression: Riemannian Geometry Proof 301

13.2. The Case Where N is a Heisenberg Group 303 13.3. The Chevalley-Vinberg Decomposition 309

13.3A. Digression: Chevalley Decompositions 309 13.3B. Weakly Commutative Spaces 314 13.3C. Weakly Commutative Nilmanifolds 317 13.3D. Vinberg's Decomposition 318

13.4. Irreducible Commutative Nilmanifolds 319 13.4A. The Irreducible Case — Classification 320 13.4B. The Irreducible Case — Structure 321 13.4C. Decomposition into Irreducible Factors 326 13.4D. A Restricted Classification 327

Chapter 14. Analysis on Commutative Nilmanifolds 329 14.1. Kirillov Theory 329 14.2. Moore-Wolf Theory 330 14.3. The Case where N is a (very) Generalized Heisenberg Group 335 14.4. Specialization to Commutative Nilmanifolds 338 14.5. Spherical Functions 341

14.5A. General Setting for Semidirect Products N x K 342 14.5B. The Commutative Nilmanifold Case 342

Chapter 15. Classification of Commutative Spaces 345 15.1. The Classification Criterion 345 15.2. Trees and Forests 350

15.2A. Trees and Triples 350 15.2B. The Mixed Case 351 15.2C. The Nilmanifold Case 353

15.3. Centers 354

15.4. Weakly Symmetric Spaces 357

Bibliography 367

Subject Index 373

Symbol Index 383

Table Index 387

Introduction

Commutative space theory is a common generalization of the theories of com­pact topological groups, locally compact abelian groups, riemannian symmetric spaces and multiply transitive transformation groups. This is an elegant meeting ground for group theory, harmonic analysis and differential geometry, and it even has some points of contact with number theory and mathematical physics. It is fascinating to see the interplay between these areas, as illustrated by an abundance of interesting examples.

There are two distinct approaches to the theory of commutative spaces: ana­lytic and geometric. The geometric approach, which is the theory of weakly sym­metric spaces, is quite beautiful, but slightly less general and is still in a state of rapid development. The analytic approach, which is harmonic analysis of commu­tative spaces, has reached a certain plateau, so it is an appropriate moment for a monograph with that emphasis. That is what I tried to do here.

Commutative pairs (G, K) (or commutative spaces G/K) can be characterized in several ways. One is that the action of G on L2(G/K) is multiplicity-free. Another is that the (convolution) algebra L1(K\G/K) of if-bi-invariant functions on G is commutative. A third, applicable to the case where G is a Lie group, is that the algebra D(G, K) of G-invariant differential operators on G/K is commutative. The common ground and basic tool is the notion of spherical function. In the Lie group case the spherical functions are the (normalized) joint eigenfunctions of the commutative algebra D(G, K). The result is a spherical transform, which reduces to the ordinary Fourier transform when G = Rn and K is trivial, an inversion formula for that transform, and a resulting decomposition of the G-module L2 {G/K) into irreducible representation spaces for G. In many cases this can be made quite explicit. But in many others that has not yet been done.

This monograph is divided into four parts. The first two are introductory and should be accessible to most first year graduate students. The third takes a bit of analytic sophistication but, again, should be reasonably accessible. The fourth describes recent results and in intended for mathematicians beginning their research careers as well as mathematicians interested in seeing just how far one can go with this unified view of algebra, geometry and analysis.

Part 1, "General Theory of Topological Groups", is meant as an introduction to the subject. It contains a large number of examples, most of which are used in the sequel. These examples include all the standard semisimple linear Lie groups, the Heisenberg groups, and the adele groups. The high point of Part 1, beside

xi i i

xiv INTRODUCTION

the examples, is construction of Haar measure and the invariant integral, and the discussion of convolution product and the Lebesgue spaces.

Part 2, "Representation Theory and Compact Groups", also provides back­ground, but at a slightly higher level. It contains a discussion of the Mackey Little-Group method and its application to Heisenberg groups, and a proof of the Peter-Weyl Theorem. It also contains a discussion of the Cart an highest weight theory with applications to the Borel-Weil Theorem and to recent results on in­variant function algebras. Part 2 ends with a discussion of the action of a locally compact group G on L2(G/T), where Y is a co-compact discrete subgroup.

Part 3, "Introduction to Commutative Spaces", is a fairly complete introduc­tion, describing the theory up to its resurgence. That resurgence began slowly in the 1980's and became rapid in the 1990's. After the definitions and a num­ber of examples, we introduce spherical functions in general and positive definite ones in particular, including the unitary representation associated to a positive definite spherical function. The application to harmonic analysis on G/K consists of a discussion of the spherical transform, Bochner's theorem, the inverse spher­ical transform, the Plancherel theorem, and uncertainty principles. Part 3 ends with a treatment of harmonic analysis on locally compact abelian groups from the viewpoint of commutative spaces.

Part 4, "Structure and Analysis for Commutative Spaces", starts with rie-mannian symmetric space theory as a sort of role model, and then goes into recent research on commutative spaces oriented toward similar structural and analytical results. The structure and classification theory for commutative pairs (G,K), G reductive, includes the information that (G, K) is commutative if and only if it is weakly symmetric, and this is equivalent to the condition that (GC,KC) is spher­ical. Except in special cases the problem of determining the spherical functions, for these reductive commutative spaces, remains open. The structure and classi­fication theory for commutative pairs (G, K), where G is the semidirect product of its nilradical N with the compact group K, is also complete, and in most cases here the theory of square integrable representations of nilpotent Lie groups leads to information on the spherical functions. The structure and classification in gen­eral depends on the results for the reductive and the nilmanifold cases; it consists of methods for starting with a short list of pairs (G, K) and constructing all the others. Finally there is a discussion of just which commutative pairs are weakly symmetric.

At this point I should point out two areas that are not treated here. The first, already mentioned, is the general theory of weakly symmetric spaces, and the closely related areas of geodesic orbit spaces and naturally reductive riemannian homogeneous spaces. That beautiful topic, touched momentarily in Section 13.1C, has an extensive literature.

The second area not treated here consists of certain extensions of (at least parts of) the theory of commutative spaces. This includes the extensive but somewhat technical theory of semisimple symmetric spaces, (the pseudo-riemannian analogs of riemannian symmetric spaces of noncompact type), the theory of generalized Gelfand pairs (G,H), and the study of irreducible unitary representations of G that have an iif-fixed distribution vector. It also includes several approaches to

NOTATIONAL C O N V E N T I O N S xv

infinite dimensional analogs of Gelfand pairs. That elegant area is extremely active but its level of technicality takes it out of the scope of this book.

Acknowledgment s

Much of the material in Parts 1, 2 and 3 was the subject of courses I taught at the University of California, Berkeley, over a period of years. Questions, comments and suggestions from participants in those courses certainly improved the exposi­tion. Some of the material in Part 3 relies on earlier treatments of J. Dieudonne [Di] and J. Faraut [Fa], and much of the material in Part 4 depends on O. Yakimova's doctoral dissertation [Y3]. In addition, a number of mathematicians looked at early versions of this book and made useful suggestions. These include D. Akhiezer (com­munications concerning his work with E. B. Vinberg on weakly symmetric spaces), D. Bao (discussions on Finsler manifolds), R. Goodman (advice on how to organize a book), I. A. Latypov and V. M. Gichev (communications concerning their work on invariant function algebras), J. Lauret, H. Nguyen and G. Olafsson (for going over the manuscript), G. Ratcliff and C. Benson (communications concerning their work with J. Jenkins on spherical functions for commutative Heisenberg nilmani-folds), and the three mathematicians who refereed this volume (for some very useful remarks).

I especially want to thank O. Yakimova for a number of helpful conversations concerning her work and E. B. Vinberg's work on classification of smooth commu­tative spaces.

Notational Conventions

M, C, M and O denote the real, complex, quaternionic and octonionic number systems. If F is one of them, then x H-> X* denotes the conjugation of F over R, F m x n denotes the space o f m x n matrices over F, and if x G F m x n then x* e F n X m

is its conjugate transpose. We write R e F n x n for the hermitian (x = x*) elements of F n x n and ReFp X n for those of trace 0, and we write I m F n X n for the skew-hermitian (x + #* = 0) elements of F n X n ; that corresponds to the case n = 1.

In general we use upper case roman letters for groups, and when possible we use the corresponding lower case letters for their elements. If G is a Lie group then g denotes its Lie algebra. If I) is a Lie subalgebra of g then (unless it is defined differently) H is the corresponding analytic subgroup of G.

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Subject Index

absolutely irreducible representation, 321 adele group, 52 adele ring, 52 adjoint representation, 121 affme group, 15 Akhiezer-Vinberg Theorem, 281 algebra

: Jordan, 27 :: euclidean, 27 :: exceptional, 27 :: formally real, 27 :: reduced structure group, 27 :: special, 27 :: structure group, 27

: Lie, 21, 88, 120 : associative, 26

almost periodic function, 217 almost-complex manifold, 133 almost-complex structure, 133, 229 a-representation, 85 amenable group, 218 antiholomorphic tangent bundle, 133 antiholomorphic tangent space, 133 approximate identity, 50, 155 associated vector bundle, 117 associative algebra, 26 automorphism group, 26

Baker-Campbell-Hausdorff formula, 88 Banach *-algebra, 56 Banach algebra

: commutative, 179 Banach-Steinhaus Theorem, 161 Bessel function, 244, 307 bilinear form

: antisymmetric, 68 : degenerate, 25 : invariant, 68 : symmetric, 68

block matrix, 13 Bochner space for G/K, 195 Bochner space for K\G/K, 187 Bochner Theorem, 187 Bochner-Godement Theorem, 187 Bohr compactification of G, 216 Borel map, 84

Borel subalgebra, 126 : real, 127

Borel subgroup, 126 : real, 127

Borel-Weil Theorem, 135 Borel-Weil-Bott Theorem, 135 bounded (G, K)—spherical function, 166,

184, 242 bounded symmetric domain

: complex, 232 : quaternionic, 232 : real, 232

Bourbaki numbering of the simple roots, 125

Bruhat decomposition, 248 bundle

: fiber :: associated, 81 :: principal, 81

: holomorphic line, 88 : structure group, 81 : vector

:: associated, 81, 117

c-function, 256 : formula, for symmetric space of

noncompact type, 257 Carcano's Theorem, 303 Cartan duality

: of orthogonal involutive Lie algebras, 228

: of riemannian symmetric spaces, 229 Cartan involution, 20, 21, 171

: conjugacy, 171 Cartan subalgebra, 122, 311 Cartan subgroup, 122

: compact part, 171 : fundamental, 171 : maximally compact, 171 : maximally noncompact, 171 : maximally split, 171 : noncompact (split) part, 171

Cartan's Highest Weight Theorems, 129 Cartan-Helgason Theorem, 132, 247 Cartan-Killing form, 121 categorical quotient, 238, 242

373

374 SUBJECT INDEX

Cauchy-Kowalevski Theorem, 133 Cauchy-Riemann equations, 133, 134 Cayley division algebra, 26 central character of a representation, 129 central reduction of a commutative pair,

320 central reduction of a Gelfand pair, 320 character (of a representation), 97

: central, 129 : infinitesimal, 334

Chebyschev polynomial, 176 Chevalley decomposition, 309

: extremely weak, 312 : very weak, 310 : weak, 310

co—isotropic subspace, 314 co—isotropic symplectic action, 314 co-rank of a group action, 317 cocycle representation, 85 coefficient homomorphism, 84 coefficient of a representation, 94 commutative Banach algebra, 179 commutative nilmanifold, 319

: irreducible, 320 commutative pair, 153

: central reduction, 320, 345 : criterion, 345 : decomposable, 345 : indecomposable, 346 : maximal, 320, 345 : principal, 327, 346 : restricted classification, 327 : Sp(l)-saturated, 327 : Sp(l)-saturated Gelfand pair, 346

commutative space, 153 commuting algebra, 61, 71 compactly embedded subgroup, 279 compatible root order, 246, 253 completely reducible representation, 60 complex manifold, 133 complex orthogonal group, 21 complex structure, 133

: integrable, 229 : invariant, 229

complex symmetric space, 229 complex symplectic group, 22 complexification, 30 complexified tangent space, 133 complexity of a group action, 317 concentrated, a measure on a set, 192 convolution

: classical, 47 : of (finite Borel) measures, 51 : on discrete group, 47 : on locally compact group, 47

convolution algebra, 49, 153 convolution of measures, 155 C*-algebra of (G, K), 188

C*-algebra of G, 59 cyclic representation, 166 cyclic vector, 166

decomposable Gelfand pair, 345 defect of a group action, 317 degree (of a vertex in a graph), 350 differentiability

: level, 119 differential operator

: invariant, 160 direct integral

: of Hilbert spaces, 75 : L2

:: of Hilbert spaces, 75 : LP

:: of Hilbert spaces, 76 :: of linear operators, 76 :: of representations, 76

direct product group, 12 discrete series

: relative, 331 distributions on manifolds, 161 doubly transitive group, 174 dual homomorphism, 211 dual lattice, 208 Dynkin diagram, 125

Engel subalgebra, 311 e-concent rated, 199 equivalence (of representations), 61 euclidean group, 15

: proper, 15 exponential map

: riemannian, 225 exponential series, 120 extension

: of a representation to its stabilizer, 83 extension property, 83 extremely weak Chevalley decomposition,

312

field : topological, 32 : p-adic, 32

Finsler geometry, 260 Finsler space, 261

: Berwald, 262 : absolutely homogeneous, 261 : distance, 261 : geodesic, 261 : homogeneous, 261 : isometry, 261 : reversible, 261 : symmetric, 261

Finsler structure, 260 Finsler symmetric space, 261

: geodesic symmetry, 261 Fock space, 89

SUBJECT INDEX 375

formal degree, 332 Fourier inversion

: compact abelian group, 213 : compact group

:: scalar, 102 : discrete abelian group, 213 : for (G, K)

:: scalar, 196 :: vector, 196

: for a locally compact abelian group, 212 : for symmetric space of noncompact

type, 258 : product of abelian groups, 213

Fourier transform : adjoint, 214 : classical, 208 : compact group

:: operator—valued, 104 : for G/K

:: vector, 196 : for K\G/K, 193 : for a locally compact abelian group,

208, 209 : for symmetric space of noncompact

type, 258 : inverse

:: classical, 208 : locally compact abelian group, 214

Freudenthal multiplicity formula, 131 Frobenius Reciprocity Theorem, 116 function

: (G, K)-spherical, 176 : spherical, 157

function algebra, 136 : antisymmetric, 136 : rotation-invariant, 136 : self adjoint, 136 : skew adjoint, 136 : symmetric, 136

functional equation (for spherical functions), 158

fundamental set : for action of a discrete group, 141 : neighbor, 141 : normal, 141

T function, 238 Gelfand pair, 153

: central reduction, 320, 345 : criterion, 345 : decomposable, 345 : indecomposable, 346 : maximal, 320, 345 : principal, 327, 346 : restricted classification, 327 : special class, 154 : Sp(l)-saturated, 327 : 5p( l ) -saturated Gelfand pair, 346

Gelfand transform, 179, 183 Gelfand-Godement-Helgason Theorem, 160 Gelfand-Mazur Theorem, 180 general linear group, 19 generalized Heisenberg group, 34 geodesic, 225 geodesic orbit space, 302 geodesic symmetry, 226 Gichev-Latypov Theorem, 137 Gindikin-Karpelevic formula, 257 (G,K) spherical functions on E n , 236 (G, K)-spherical function, 176 (G, K)-spherical function

: Harish-Chandra formula, 254 : on symmetric space of noncompact

type, 254 global inner product, 78 Grassmann manifold, 29

: complex, 232 : oriented real, 232 : quaternionic, 232 : real, 232

group, 3 : Heisenberg, 33, 87 : Heisenberg for F n , 34 : Lie, 119 : adele, 52 : affine, 15 : algebra, 49 : automorphism, 26 : complex orthogonal, 21 : complex symplectic, 22 : direct product, 12 : euclidean, 15 : general linear, 19 : generalized Heisenberg for ¥p,q, 34 : indefinite orthogonal, 22, 23 : indefinite special orthogonal, 24 : indefinite special unitary, 24 : indefinite symplectic unitary, 23 : indefinite unitary, 23 : linear Lie, 20 : linear algebraic, 20 : ordinary orthogonal, 22 : orthogonal, 21, 23 : projective general linear, 26 : proper euclidean, 15 : real orthogonal, 22 : real symplectic, 22 : semidirect product, 14 : special linear, 20 : special orthogonal, 22, 24 : special unitary, 23, 24 : symplectic, 21 : symplectic unitary, 23 : topological, 3 : topological quotient, 4 : unitary, 23

376 SUBJECT INDEX

:: projective, 85 : very generalized Heisenberg for

j r5x(t ,u) ? 34

: volume preserving, 20 group algebra, 49

Haar integral, 35 : left, 35 : right, 36

Haar measure, 35 : left, 35 : normalized, 93 : right, 36

Hahn-Banach Theorem, 162 has square integrable representations, 331 Heisenberg commutation relations, 88

: uniqueness of, 88 Heisenberg group, 33, 87, 303 Heisenberg group over C, 34 Heisenberg group over F, 34 Hermite monomial, 89 Hermite polynomial, 88 hermitian symmetric space, 229 highest weight of a representation, 129 Hilbert-Schmidt

: inner product, 100, 147 : norm, 100 : operator, 100, 147

Hirzebruch Proportionality Principle, 141 holomorphic function, 133 holomorphic line bundle, 88 holomorphic tangent bundle, 133 holomorphic tangent space, 133 holomorphic vector bundle, 134 Hom(B 7 r i ,B 7 r 2 ) , 67 homogeneous space, 9 homogeneous tree, 175 hypergeometric equation, 250, 255 hypergeometric function, 251, 255 hypergroup, 51, 202

ideal : regular, 179

identity component, 6 indecomposable

: algebraically, 60 : topologically, 60

indecomposable Gelfand pair, 346 indefinite orthogonal group, 22, 23 indefinite special orthogonal group, 24 indefinite special unitary group, 24 indefinite symplectic unitary group, 23 indefinite unitary group, 23 index (or a vertex of a tree), 175 indicator (= characteristic) function, 199 indivisible restricted root, 246, 253 induced spherical function, 169 induction by stages, 79

infinitesimal character (of a representation), 334

infinitesimal transvection, 226 inner product

: global, 78 integrability condition (C°°), 133 integrability condition (C w ) , 133 integrable complex structure, 229 integration on homogeneous space, 45 intertwining operator, 61, 67 invariant integral, 35 invariant metric, 15 invariant vector field, 119 inverse Fourier transform, 208 Inverse Function Theorem, 120 inverse spherical transform

: for (G,K), 191 irreducible

: algebraically, 59 : topologically, 59

isometry group, 225 isotropic subspace, 314 isotropy subgroup, 9, 226 Iwasawa decomposition, 171, 248

Jacobi function, 256 Jacobi identity, 120 Jacobi polynomial, 251, 252 Jacobson—Morosov Theorem, 127 joint T>(G, K)-eigenfunctions, 160 Jordan algebra, 27

: euclidean, 27 :: table, 28

: exceptional, 27 : formally real, 27 : reduced structure group, 27 : special, 27 : structure group, 27

X-fixed vector, 167 Killing form, 121 Kimmelfeld-Vinberg Theorem, 274 Kirillov construction, 330 Kostant multiplicity formula, 131 Kramer classification of spherical spaces,

282

Laguerre polynomial, 307 Leray spectral sequence, 135 level of a weight, 131 level of differentiability, 119 Levi component, 309 Levi decomposition, 300 Levi subalgebra, 300, 309 Lie algebra, 21, 88, 120

: Cartan decomposition, 122 : Cartan subalgebra, 122 : Dynkin diagram, 125 : Schlafli-Dynkin diagram, 125

SUBJECT INDEX 377

Weyl group, 124 center, 121 centralizer, 122 commutative, 121 direct sum, 120 exponential map, 120 homomorphism, 120, 121 ideal, 120 nilpotent, 121 normalizer, 122 of a Lie group, 120 orthogonal involutive, 226 radical, 121 rank, 124 reductive, 121 root :: Weyl chamber, 124 :: chain, 124 :: hyperplane, 124 :: length, 124

root decomposition, 122 root lattice, 132 root length, 125 root reflections, 124 root system, 122

positive, 123 rank, 124 simple, 123

roots, 122 semidirect sum, 120 semisimple, 121 simple, 121 solvable, 121 solvable radical, 121 splittable, 310 subalgebra, 120

Lie group, 20, 119 : Cartan subgroup, 122 : Lie subgroup, 120 : Weyl group, 124 : centralizer, 122 : exponential map, 120 : homomorphism, 121 : normalizer, 122 : rank, 124

Lindelof, 142 linear algebraic group, 20, 272

complex, 272 real, 272 reductive, 272

linear algebraic groups, 82 linear functional

: multiplicative, 156, 179 linear Lie group, 20 LP S-bandlimited, 202 Lp e-concentrated, 202 Lp-induced spherical function, 169 Lp 5-concentrated, 202

Mackey Little Group Theorem, 83 Mackey obstruction, 86 Mal'cev splitting of a Lie algebra, 310 matrix coefficient, 94 maximal commutative pair, 320 maximal compact subgroup, 171

: conjugacy, 171 maximal Gelfand pair, 320, 345 maximal ideal space, 179

: topology, 182 : weak * topology, 182

maximal weight of a representation, 129 mean (on a topological group), 218 measurable set, 75 measure, 75

: Plancherel :: for (G,K), 191

: Radon, 157 : atomic, 104 : spectral, 218 : spherical, 156

measure algebra, 50, 51 measure space, 75

complete, 75 finite, 93 product, 93

metaplectic representation, 91 minimal orthogonal involutive Lie algebra,

227 minimal parabolic subalgebra, 171 Minkowski norm, 260 modular function, 41 module (of an automorphism), 41 multiplication of sets, 155 multiplicative linear functional, 156, 179 multiplicity free, 197 multiplicity free vs. "multiplicity free", 307 multiplicity of a subrepresentation, 70 multiplicity of a weight, 128

Nelson's Theorem, 162 : Garding's proof, 162

Newlander-Nirenberg Theorem, 133 nilradical or nilpotent radical

: of a Lie algebra, 300 : of a Lie group, 300

norm : global

:: L°°, 78 :: LP, 78

normalized character (of a representation), 97

octonion division algebra, 26 automorphism group, 27 multiplication diagram, 27 multiplication table, 26

octonion hyperbolic plane, 233

378 SUBJECT INDEX

octonion projective plane, 233 1-parameter subgroup, 119 one parameter subgroup, 119 operator

: Hilbert-Schmidt, 147 : compact, 72 : completely continuous, 72 : trace class, 147

orbit, 9 ordinary orthogonal group, 22 orthogonal group, 21, 23 orthogonal involutive Lie algebra, 226

: Cartan duality, 228 : compact group, 228 : compact type, 228 : direct sum, 227 : euclidean, 227 : four classes of irreducible, 228 : irreducible, 227 : isomorphism, 227 : minimal, 227 : noncompact type, 228

oscillator representation, 91

p-adic integers, 51 p-adic number field, 32 Panyushev Theorems, 276 parabolic subalgebra, 126

: minimal, 171 : real, 127

parabolic subgroup, 36, 126 : real, 127

parallel tensor field, 229 Peter-Weyl Theorem, 99 PfafRan

: of an antisymmetric bilinear form, 333 : polynomial, 333

Plancherel density : for symmetric space of noncompact

type, 258 Plancherel formula

: compact group :: Hilbert-Schmidt, 101 :: operator-valued, 104 :: trace form, 102

: for G/K, 196 : for K\G/K, 193 : locally compact abelian group, 214

Plancherel measure : for (G,K), 191

point mass, 155 Poisson algebra, 314 Poisson bracket, 314 polarization

: real, 329 polonais

: group, 84 : space, 84

Pontrjagin Duality Theorem, 214 positive definite (G, K)-spherical function,

167, 184 positive definite function, 165

: spherical, 167 positive restricted root system, 171 positive Weyl chamber, 254 primary constituents of a representation, 71 primary decomposition of a representation,

71 primary representation, 71 primary subrepresentation, 71 primary subspace (of a representation

space), 71 principal fiber bundle, 81 principal Gelfand pair, 327, 346 principal series representation, 174 principal triple (F, F , V), 351 product

: direct, 12 : semidirect, 14

projective general linear group, 26 projective kernel, 140 projective space, 28 projective unitary group, 85 projective unitary representation, 85

quasi-character, 207 quaternion

: structure :: invariant, 113

: structure on a vector space, 113 quaternionic structure, 230 quotient representation, 60

radial part of the Laplace—Beltrami operator, 254

radical : nilpotent, 300 : solvable, 300 : unipotent, 309

Radon measure, 35, 157 rank

: of a riemannian symmetric space, 229 : real, of a semisimple Lie group, 229

rank of a group action, 317 real form, 30

: of a complex representation, 113 : of a complex vector space, 113

real orthogonal group, 22 real polarization, 329 real rank, 229 real symplectic group, 22 reductive component, 309 reductive subalgebra, 309 regular ideal, 179 regular set, 130 relative discrete series representation, 331

SUBJECT INDEX 379

representation : Banach algebra on a Banach space, 57

:: bounded, 57 : Segal-Shale-Weil, 91 : Weil, 91 : absolutely irreducible, 321 : adjoint, 121 : admissible, 148 : algebraic direct sum, 65 : basic weight, 131 : character, 148 : cocycle, 85 : compact, 72 : completely continuous, 72 : completely reducible, 60 : complexification of real, 113 : contragredient, 64, 329 : distribution character, 148 : dual, 64, 329 : equivalence, 61 : finite dimensional

:: character, 97 :: normalized character, 97

: finite multiplicity, 71 : fundamental weight, 131 : global character, 148 : group on Banach space, 56

:: bounded, 56 : induced

:: by stages, 79 :: LP, 78 unitary, 78

: infinitesimal character, 334 : left regular

:: of group, 56 :: of group algebra, 57 :: of measure algebra, 57 :: on LP{G), 56

: linear, 85 : metaplectic, 91 : multiplicity-free, 71 : norm—preserving, 56 : orthogonal direct sum, 65 : oscillator, 91 : principal series, 174 : quaternionic, 113 : quotient representation, 60 : real, 113 : relative (mod Z) discrete series, 331 : right regular

:: of group, 57 :: on LP(G), 57

: semisimple, 60 : spherical principal series, 174 : square integrable (mod Z), 331 : subquotient, 60 : subrepresentation, 60 : tempered, 55

: topologically completely irreducible, 62 : unitary, 56

:: projective, 85 : unitary equivalence, 61 : unitary principal series, 174 : weight, 128 : weight lattice, 131 : weight space decomposition, 128 : L°° discrete direct sum, 65 : Lp direct sum, 64 : Lp discrete direct sum, 65

representation space, 56 representative function, 140 restricted root space decomposition, 246,

253 restricted root system, 171, 253 restricted Weyl group, 254 Riemann-Lebesgue Lemma, 183, 186 riemannian covering, 229 riemannian homogeneous space, 226 riemannian nilmanifold, 301 riemannian symmetric space, 226

: for an orthogonal involutive Lie algebra, 227

: rank, 229 Riesz-Thorin interpolation, 200 root lattice, 132

scalar Fourier inversion : for (G,K)t 196

scalar Fourier transform, 193 scalar part, 230 Schlafli-Dynkin diagram, 125 Schur Orthogonality Relation, 96 Schur's Lemma, 61 Segal-Shale-Weil representation, 91 semidirect product group, 14 seminorm, 161 semisimple representation, 60 <T—compact, 11 simple restricted root system, 171 skew-gradient, 315 Sobolev Inequalities, 162 solvable radical

: of a Lie algebra, 300 : of a Lie group, 300

Sp(l)-saturated Gelfand pair, 327, 346 special linear group, 20 special orthogonal group, 22, 24 special unitary group, 23, 24 spectral measure, 218 spectral radius, 180 Spectral Radius Theorem, 181 Spectral Theorem, 219 spectrum (of an element of a Banach

algebra), 180 spherical

: homogeneous space

380 SUBJECT INDEX

:: complex, 272 :: real, 272

: pair :: complex, 272 :: real, 272

: subgroup, 272 (G, K)-spherical functions, 184, 242 spherical function, 157

: bounded, 166 : of type <5A, 270 : positive definite, 167

spherical function for (G, K) : positive definite, 184

spherical functions for (G,K), 184, 242 : bounded, 184

spherical harmonics, 137 spherical inversion

: for symmetric space of noncompact type, 258

spherical measure, 156 spherical principal series representation,

174 spherical transform, 184

: for K\G/K, 193 : inverse

:: for (G,K), 191 splittable Lie algebra, 310 stabilizer

: of a representation of a subgroup, 83 Stone's Theorem, 219 Stone-Weierstrass Theorem, 136 subalgebra

: Borel, 126 :: real, 127

: Cartan, 122, 311 : Engel, 311 : parabolic, 126

:: real, 127 : reductive, 309 : reductive in the large algebra, 309

subgroup : Borel, 126

:: real, 127 : Cartan, 122 : isotropy, 226 : one-parameter, 119 : parabolic, 126

:: real, 127 submanifold

: regularly embedded, 120 subquotient representation, 60 subrepresentation, 60 symmetric algebra, 334 symmetric space

: grassmannian, 232 : hermitian, 229 : quaternionic, 230 : riemannian, 226

symmetry : geodesic, 226

symplectic action, 314 : co—isotropic, 314

symplectic group, 21 symplectic manifold, 314 symplectic unitary group, 23

tangent space, 225 tensor power

: antisymmetric :: of Banach representations, 67 :: of Banach spaces, 67

: of Banach representations :: projective, 67

: of Banach spaces :: projective, 67

: symmetric :: of Banach representations, 67 :: of Banach spaces, 67

tensor product : of Banach algebras

:: projective, 68 : of Banach representations

:: exterior projective, 66 : of Banach spaces

:: algebraic, 65 :: projective, 66

: of Hilbert spaces :: projective, 67

: of unitary representations :: exterior projective, 67 :: interior projective, 67

Thomas' Theorem, 160 Titchmarch Inequality, 200 topological action, 8 topological field, 32 topological group isomorphism, 11 topological space

: regular, 3 topological transformation group, 8 topology

: quotient, 4 : subspace, 4

totally real submanifold, 239 trace class

: operator, 100, 147 translation, 3

: left, 3 : on quotient space, 5 : right, 3

translation—invariant vector field, 119 transvection, 226 tree, 175, 350

: homogeneous, 175 : rooted, 350

:: root, 350 trigonometric polynomial, 217

SUBJECT INDEX 381

2-step Nilpotent Theorem, 299 : structure of commutative spaces, 264 Two-Step Nilpotent Theorem, 299 Type I, 82

uncertainty principle : classical, 199 : for G/K, 204 : for K\G/K, 202 : for locally compact abelian groups, 199

unimodular (group), 41 unipotent radical, 309 unitary dual, 70 unitary dual group, 208 unitary equivalence (of representations), 61 unitary group, 23 unitary principal series representation, 174 universal covering group, 8 universal covering space, 7 universal enveloping algebra, 334

vector field : invariant, 119

vector Fourier inversion : for (G,K), 196

very generalized Heisenberg group, 34 Vinberg's Decomposition Theorem, 318 volume preserving group, 20

weak containment, 55 weak symmetry

: conditions on, 267 : differential—geometric, 266 : group-theoretic, 265

weakly commutative coset space, 314 weakly commutative pair, 314 weakly symmetric

: complex pair (GC,HC), 276 :: compact real form of, 276 :: real form of, 276 :: weak symmetry of, 276

: coset space G/K, 265 : pair (G ,K) , 265 : riemannian manifold, 264 : weak symmetry, 265

weight : highest, 129 : maximal, 129 : multiplicity, 128 : of a representation, 128 : space decomposition of representation

space, 128 weight lattice, 131

: positive, 132 Weil representation, 91 Weyl character formula, 130 Weyl degree formula, 131 Weyl group, 124 Weyl involution, 277, 358 working assumptions

Symbol Index

A = expG(a) , 245 A(R), range of the spherical transform, 191 A : G = NAK -+ a by

g = v(g)expA(g)n(g), 254 .A(7r), commuting algebra of 7r, 71 a, maximal abelian subspace of s in Cartan

decomposition gg = t + s, 245 Ad, adjoint representation of Lie group, 121 ad = dAd, adjoint representation of Lie

algebra, 121 ag, conjugation, 3 AP(G), almost periodic functions on G,

217 Aut(Hn), automorphism group of Hn, 87,

303

B(G/K), Bochner space for G/K, 195 B(K\G/K), Bochner space for K\G/K,

187 Bq(G; A), Borel g-coboundaries, 84 Bn, representation space of 7r, 56 B(B), algebra of bounded operators on

Banach space B, 56 &/(€,»?) = / ( K , 17]), 329 BS = BS(G,K), bounded (G, K)-spherical

functions, 184

C(K\G/K), K-bi-invariant continuous functions on G, 153

Ct{G) = {fe CC{G) I f(G) C M=0}, 37 C w , real analytic, 119 C-°° (G) , distributions on G, 161 C-°°IG/K), distributions on G/K, 161 C-°°(K\G/K), distributions on K\G/K,

161 CC(G), continuous compactly supported

functions G -+ C, 37 CC(K\G/K), K-bi-invariant continuous

compactly supported functions on G, 153

Cc~°°(G), compactly supported distributions on G, 161

C^°°(G/K), compactly supported distributions on G/K, 161

C^~°°(K\G/K), compactly supported distributions on K\G/K, 161

C, complex number field, xv C m X n , m x n complex matrices, xv C(K\G/K), if-bi-invariant continuous

functions that vanish at 00 on G, 153 Coo(X), continuous functions that vanish at

infinity on X, 183 Coo(MA), 183 Xn, character of finite dimensional

representation 7r, 97 Xn(g) = trace n(g), 97 Cn//Kc, categorical quotient, 238 C*(G), G*-algebra of G, 59 C*(G, K), the G*-algebra of (G, K), 188

P(G) , left-invariant differential operators on G, 160

V(G,K), G-invariant differential operators on G/K, 160

deg(7r) = dimi^Tr, degree of the representation 7r, 96

deg(7r), formal degree of relative discrete series representation 7r, 332

A G , modular function of G, 41 AG/H(h) = AG(h)/AH(h),U Der(Q, Lie algebra of derivations of [, 120 dist, distance function on V(T), 175

E(T), edges of tree T, 175, 350 ^6,F 4 , collineation group of octonion

projective plane, 233 exp^ : TX(M) —+ M, riemannian

exponential map, 225

[^r(/)](a;) = ^uj(f)uuj G HUJ, vector Fourier transform, 196

¥p,q, p x q matrices over F with hermitian form of signature (p, q), 33

/ M = fGf(9)UJ(9~1)diJ,G(g) = m w ( / ) , spherical transform, 184

f(n) = trace 7r(/), scalar-valued Fourier transform, 102

/ H^ JF(/) = (7r(/)) [7r ]Gg, operator-valued Fourier transform, 104

/°(P) = / ( P " 1 ) , 1 5 4 fHg) = IK IK f(ki9k2) dp,K {k1)dfiK (fc2),

153

384 SYMBOL INDEX

f(g) = 1(0(9)), 154 fu,v, matrix coefficient, 94 2F1, hypergeometric function, 251, 255 F4/Spin(9), octonion projective plane, 233 ^4^4/Spin(9), octonion hyperbolic plane,

233

G —* G/H, projection to quotient, 121 G°, topological component of 1 EG, 6 G ^ ) , Borel g-cochains, 84 G*A, adele group of GJK, 52 G c , complexification of G, 30 Gfc n (C) , complex Grassmann manifold, 29 Gfc n (H) , quaternion Grassmann manifold,

' 29 Gfc>n(R), real Grassmann manifold, 29 GpjqjF = #p,q;F X ^ ( p , ^ ) , 34 (25, universal enveloping algebra of Q, 164

G = G^, Bohr compactification of G, 216 G, unitary dual of G, 70 G, universal covering group of G, 8 Q, Gelfand transform, 183 (g, <J, 6), orthogonal involutive Lie algebra,

226 5 = £ + m, decomposition by the symmetry,

226 0, Lie algebra of G, 21 g — t + s, Cartan decomposition, 245 GL / (V) ,GL / (n ;F) , preserves volume, 20 GL(B), bounded operators on B with

bounded inverse, 56 GL(V), GL(n;F) , general linear group, 19 Gmin, subgroup of I (M) generated by

transvections, 229 Qmin = [tn, m] + m, minimal orthogonal

involutive Lie algebra, 227

H := ZQ{\)) = 7M, maximally split Cartan subgroup of G, 245

if := ZQ(\)) =T x A, maximally split Cartan subgroup of G, 253

ii~2(M;Z), integral cohomology in degree 2, 230

H*(G;A) = Z<?(G; A)/B*(G\ A), Borel g-cohomology, 84

7J^ , if-fixed vectors in H^, 167 ifn = Hn.£, usual Heisenberg group, 34 i i n , Heisenberg group, 87, 303 H(i/>)i ^-pr imary subspace of Hn, 71 ^n;F> generalized Heisenberg group based

' o n F n , 34 ^p,q;F, generalized Heisenberg group based

' o n F ^ , 34, 303 ,t,tt;F) very generalized Heisenberg group

based on F s x ^ ' u ) , 34 H, quaternion division algebra, xv H m X n , m x n quaternion matrices, xv

7i2 = fY Hy dr(y), L2 direct integral of Hilbert spaces, 75

Hp = JY Hy dr(y), Lp direct integral of Hilbert spaces, 76

\) := t + a, maximally split Cartan subalgebra of g, 245, 253

Hom(B 7 r i ,B 7 r 2 ) , 67 Homer(Bn i , BTT2), intertwining operators,

67

I(TV,TT'), intertwining operators Bn —> Bnf, 61

jT(7ri,7T2), intertwining operators, 67 I m C n X n , complex skew-hermitian n x n,

xv I m H n X n , quaternion skew-hermit ian

n x n, xv I m O n X n , octonion skew-hermitian n x n,

xv ImIR n X n , real skew-hermitian n x n, xv IndS(C), spherical function L2-induced

from C, 169 IndQ'p(C), spherical function Lp—induced

from C, 169 Ind^(?7o), unitarily induced representation,

78 Ind^'p(?7o), Lp induced representation, 78 Int(g), group generated by Ad(exp(g)), 126 JG / ( X ) M G ( X ) ' left Haar integral on G, 35 SG/H <I>(9H) d\iG/H (gH), 45 7(M), isometry group of riemannian

manifold M, 225

Ju, Bessel function of the first kind of order 1/, 244, 307

KA, adele ring of algebraic number field K, 52

K:G = NAK -+ K by g = v(g)expA(g)K(g), 254

L*, dual lattice, 208 Lp(K\G/K), K-bi-invariant Lp functions

on G,153 £(w, S + ) , length of w in positive system

S + , 125 lg, left translation, 3 Lm , generalized Laguerre polynomial of

order n - 1, 307 Ak(B), kth antisymmetric power of Banach

space B, 67 Afc(7r), kth antisymmetric power of Banach

representation 7r, 67 A r t , root lattice of a Lie algebra, 132 Awt,G, weight lattice of a Lie group, 132 ^wt G~> Positive cone in weight lattice of G,

'132 Awt, weight lattice of a representation, 131

SYMBOL INDEX 385

A J t , positive cone in weight lattice of g, 132

A, left regular representation, 56 Xf, eigenvalue on / of convolution by CJ,

158

M = ZK(A), centralizer of A in K, 245 M(R), finite complex-valued Radon

measures on R, 189 M + ( P ) , non-negative finite Radon

measures on P , 189 A4^4, multiplicative linear functionals on

commutative Banach algebra A, maximal ideal space of A, 179

m = 3e(a), centralizer of a in t, 245 m(/) = J G / ^ M ^ " 1 ) ^ ^ ) ' 1 5 7

Meas(G), measure algebra of group G, 50 \±G, left Haar measure on G, 35 J^P, Plancherel measure for (G,K), 191 TO(^,7T), multiplicity of ^ in TT, 70

AT = exp(n), nilradical of minimal parabolic subgroup, 254

NG{H), normalizer of H in G, 122 n = $^7Gl]+(a a) 0~7> nilradical of minimal

parabolic subalgebra, 254 rig(rj), normalizer of f) in g, 122 ||0||oo, global L°° norm, 78 \\(J)\\P, global L^ norm, 78 IMIspec» spectral radius of a, 180 ||/1|00, Lebesgue sup norm, 37 v:G = NAK - • AT by

fir = i/(flf)expi4(^)«(^), 254

o(y,b),o(y), 21 0(n) = £/(n;M), orthogonal group, 23 0 ( n ; C ) , complex orthogonal group, 21 0(Pi q) — U(p, q\ M), indefinite real

orthogonal group, 23 0(Pj<l)i 0(n), real orthogonal groups, 22 O, octonion division algebra, xv, 26 O m X n , m x n octonion matrices, xv Of, (coadjoint) orbit Ad* (AT)/, 329 ujf, symplectic form on Of defined by bf,

329

P = P(G, K), positive definite (G, AT)-spherical function, 184

P(G,K), positive definite (G, AT)-spherical function, 242

P(f) = Pf(uf), Pfaffian polynomial, 333 P 2 (©) , octonion (or Cayley) projective

plane, 29 P n ( C ) , complex projective n-space, 29 P n ( H ) , quaternion projective n-space, 29 P n ( R ) , real projective n-space, 29 Pf(u;), Pfaffian of LU, 333 PGL(2;K) = GL(2;K)/(center), 176 PGL(2;D) and GL(2;D), 176

PGL(-), projective general linear group, 26 ^>a, simple restricted root system, 171 4% , Jacobi function, 256 Vxti) = fKe<iX+l')(M'"'))dnK(k),

Harish—Chandra formula for spherical function, 254

n = SY ny ^ T ( ^ ) ' Lp direct integral of representations, 76

[iTf] G A/", representation class defined by / G n*, 329

Pn,V(z) = 2 F i ( - n , i z + v + n + l,w + l; ^ ) , Jacobi polynomial, 251

ProjRep(G), projective unitary representations of G, 85

^ , simple subsystem of root system S, 123 PU(H), projective unitary group, 85

Q p , p—adic number field, 32

R = R(G, K), maximal ideal space of C*(G,K), 188

R, real number field, xv M m X n , m x n real matrices, xv rg, right translation, 3 R e C n X n , complex hermitian n x n, x.v R e F j X n , trace 0 elements of F n X r \ xv R e H n X n , quaternion hermitian n x n, x.v R e O n X n , octonion hermitian n x n, xv R e M n X n , real hermitian n x n, x.v Reprai(G), a-representations of G, 85 p, half the sum of the positive roots, 130 p, right regular representation, 57 Rn//K, categorical quotient, 242 X, semidirect product, 14

S = S(G, K), (G, AT)-spherical functions, 184

Sk(B), kth symmetric power of Banach space P , 67

Sk(7r), kth symmetric power of Banach representation 7r, 67

<S, spherical transform map, 193 S : ProjRep(G) -+ H2{G;C), Mackey

multiplier, 85 5(3), symmetric algebra of 3, 334 sgrad / , skew-gradient of / , 315 £(g, a), restricted root system, 246 S(g, J)), restricted root system, 253 £($, rj), root system, 246 £ + , positive subsystem of root system £ ,

123 £ a , restricted root system, 171 £a , positive restricted root system, 171 <j(a) = (j^(a), spectrum of a G A, 180 S k e w C m X m , 234 SL(V),SL(n;F), 20 SO(n), special orthogonal group, 24 SO{n, C), SO(p, q), SO(n), special

orthogonal groups, 22

386 SYMBOL INDEX

SO(p,q), indefinite special orthogonal group, 24

SO(r + s)/[SO(r) x SO(s)], oriented real Grassmann manifold, 232

SO(r,s)/[SO(r) x SO(s)}, real bounded domain, 232

SO*(V,s),SO*(2n), defined by quaternion skew-hermitian form, 24

Sp(V,b),Sp(V), 21 5p(m;C), complex symplectic group, 22 Sp(m;R), real symplectic group, 22 Sp(n) = £/(n; H), symplectic unitary group,

23 Sp(p,q) = U(p,q;M), indefinite symplectic

group, 23 Sp(r + s)/[Sp(r) x Sp(s)\, quaternionic

Grassmann manifold, 232 Sp(r,s)/[Sp(r) x 5p(s)], quaternionic

bounded domain, 232 SU(n), special unitary group, 24 SU(n;F), special unitary group, 23 SU(p,q), indefinite special unitary group,

24 SU(r + s)/S(U(r) x U(s)), complex

Grassmann manifold, 232 SU(r,s)/S(U(r) x U(s)), complex bounded

domain, 232 S u p p ( / ) , support of a function / , 37 sx : M —>• M, geodesic symmetry to M at

x, 226 S y m C m X m , 234

T = fYpTy dr(y)7 LP direct integral of linear operators, 76

T, (rooted) tree, 350 T0,1(X), antiholomorphic tangent bundle,

133 T 1 ' ° (X) , holomorphic tangent bundle, 133 TX(M), real tangent space to M at x, 225 TX(X)C, complexified tangent space, 133 Tx' (X), antiholomorphic tangent space,

133 2V (X), holomorphic tangent space, 133 t, Cartan subalgebra of m, 245 (T,S)HS = trace TS*, 100 TV , character of finite dimensional

representation 7r, 97 TTT(#) = (deg?r)trace n(g), 97 Tgi translation by g, 5, 8 TCI, topologically completely irreducible,

62 Trig(G), trigonometric polynomials on G,

217 tu(^y(t)) = 7(t + w), transvection along

geodesic 7, 226

C/(n) = C/(n;C), unitary group, 23 U(n;¥), unitary group over F, 20, 23

U(p,q) = U(p,q;C), indefinite complex unitary group, 23

U(p, g;C), indefinite complex unitary group, 23

V(T), vertices of tree T, 175, 350

W(g, a), restricted Weyl group, 254 Wt(r), weight system of representation r ,

128

X/A, quotient of space X by algebra A, 136

£i, basic (fundamental) weights, 131

Zi(G;A), Borel g-cocycles, 84 ZG(H), centralizer of H in G, 122 Zf, ring of integers in algebraic number

field F, 51 Z p , ring of p—adic integers, 51 3, universal enveloping algebra of 3, 334 3g(rj), centralizer of J) in 0, 122

Table Index

Adjoint Representations, Classical Groups, 130

Adjoint Representations, Exceptional Groups, 130

Benson-Jenkins-Ratcliff Classification, 305, 306

Brion-Mikityuk-Yakimova Classification : Compact, Semisimple, Non—Simple

Weakly Symmetric Spaces, 289 : Complex Semisimple Non-Simple

Weakly Symmetric Spaces, 288

Classical Simple Lie Groups, 126 Commutative Principal Pairs, 355 Comparison of Tables 13.4.1 and 13.2.5, 326 Connected Dynkin Diagrams, 125, 126

Dynkin Diagrams, 125

Hermitian Symmetric Spaces, 233 : Real Forms, 235

Kac Classification, 305, 306 Kramer Classification

: of Compact Spherical Spaces, 282 : of Complex Spherical Spaces, 283 : of Noncompact Real Spherical Spaces,

286

Maximal Indecomposable Principal Saturated Pairs, 328, 347

Maximal Irreducible Nilpotent Gelfand Pairs, 320

Maximal Principal Indecomposable Non-Reductive Non-Nilmanifold Non-5p(l) -Saturated Gelfand Pairs, 352, 353

"Multiplicity Free" Irreducible Representations, 305, 306

Octonion Division Algebra, 27

Quaternionic Symmetric Spaces, 235 : Complex Forms, 236

Rank 2 Root Systems, 124 Restricted Root System

: Compact, Real Rank 1, 250

: Hyperbolic Spaces, 255 : Noncompact, Real Rank 1, 255 : Spheres and Projective Spaces, 250

Riemannian Symmetric Spaces : G/K with G Classical Simple, 232 : G/K with G Exceptional Simple, 232 : Complex Cases, 233 : Group Manifolds and their

Noncompact Duals, 231 : Quaternionic Cases, 235 : Spaces of Rank 1, 233

Simple Formally Real Jordan Algebras, 28

Vector Representations, 129 Vinberg Classification of Maximal

Irreducible Nilpotent Gelfand Pairs, 320

Weakly Symmetric Spaces : G/K with G Compact Simple, 282 : Branching Compact Non-Semisimple

Circle Bundles, 292 : Branching Noncompact Reductive

Non-Semisimple Circle Bundles, 293 : Circle Bundles over Hermitian

Symmetric Spaces, 268 : Compact Non—Semisimple Circle

Bundles, 292 : Compact, Semisimple, Not Simple, 289 : Complex Simple Cases, 283 : Complex, Semisimple, Not Simple, 288 : Noncompact Real Simple Cases, 286 : Noncompact Reductive

Non-Semisimple Circle Bundles, 292

Yakimova Classification : Maximal Indecomposable Principal

5p( l ) -Saturated Gelfand Pairs, 348 : Maximal Indecomposable Principal

Saturated Nilpotent Gelfand Pairs, 328

: Maximal Principal Indecomposable Non-Reductive Non-Nilmanifold Non-Sp(l)-Saturated Gelfand Pairs, 352, 353

387

Titles in This Series

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Novokshenov, Painleve transcendents, 2006 127 Nikolai Chernov and Rober to Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson , and Wil l iam T. Ross , The Cauchy Transform,

2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 123 Barbara Fantechi, Lothar Gottsche , Luc Illusie, Steven L. Kle iman, Ni t in

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122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005

121 Anton Zettl , Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian M a and Shouhong Wang, Geometric theory of incompressible flows with

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TITLES IN THIS SERIES

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operators, and layer potentials, 2000 80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module

theory, 2000

For a complete list of t i t les in this series, visit t he AMS Bookstore at w w w . a m s . o r g / b o o k s t o r e / .