cohomology theory of topological transformation groups

Ergebnisse der Mathematik und ihrer Grenzgebiete Band 85
Herausgegeben von P. R. Halmos P. J. Hilton R. Remmert B. Sz6kefalvi-Nagy
Unler Mitwirkung von L. V. Ahlfors R. Baer F. L. Bauer A. Dold J. L. Doob S. Eilenberg K. W Gruenberg M. Kneser G. H. Muller M. M. Postnikov B. Segre E. Sperner
GeschiiftifUhrender Herausgeber P. J. Hilton
WuYi Hsiang
Wu Yi Hsiang
AMS Subject Classification (1970): 57 Exx
ISBN-l3: 978-3-642-66054-2 DOl: 10.1007/978-3-642-66052-8
e-ISBN-13: 978-3-642-66052-8
Library of Congress Cataloging in Publication Data. Hsiang, Wu Vi, 1937-. Cohomology theory of topological transformation groups. (Ergebnisse der Mathematik und ihrer Grenzgebiete; Bd. 85). Bibliography: p. Includes index. 1. Transformation groups. 2. Topological groups. 3. Homology theory. I. Title. II. Series. QA613.7.H85. 514'.2. 75-5530. This work is su bject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically those of translation, reprinting, re-use of illustrations, broadcasting. reproduction by photocopying machine or similar means. and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1975.
Softcover reprint of the hardcover 1 st edition 1975
Introduction
Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tlp by more general compact groups. It is usually rather straightforward to deduce similar fixed point theorems for actions of p-primary groups (or extensions of torus groups by p-primary groups) directly from the corresponding fixed point theorems for actions of the group 7l p • However, various efforts to extend such fixed point theorems beyond p-pi·imary groups (or extensions of torus by p-primary groups) all eventually wound up with puz zling counter-examples [C 6, C 8, F 2, H 5]. On the other hand, if the group is an elementary p-group, (i e., 7l; or T'" if p =0), then a far-reaching generalization of the Smith fixed point theorem, valid for all finite dimensional, locally compact spaces, can be formulated and proved in the framework of equivariant cohomology. (cf. Theorem (IV. i), § i, Ch. IV).
The basic setting for our approach using the cohomology theory in compact topological transformation groups is the following equivariant cohomology theory introduced by A. Borel [B 5]. Let G be a compact Lie group and let X be a given G-space. Then the equivariant cohomology of the G-space X is defined to be the ordinary cohomology of the total space XG of the universal bundle, X -+ XG -+ BG, with the given G-space as the typical fibre. The reasons for adopting such an equivariant cohomology theory in terms of the universal bundle con struction are roughly the following:
(i) Intuitively and heuristically, the complexity of the G-action on X will be reflected in the complexity of the associated universal bundle, X -+ XG -+ BG; and the classical characteristic class theory clearly demonstrates that cohomology
VI Introd uction
theory can then be used to detect the complexity of the bundle, which, in turn, also detects the complexity of the· G-action on X itself. Therefore, the above definition of equivariant cohomology simply formalizes and also generalizes the classical characteristic class theory to the study of the topology of general fibre bundles.
(ii) From a technical standpoint, the above equivariant cohomology theory naturally and successfully brings together the modern theories of fibre bundles, spectral sequences and sheaves in a nice convenient way. Hence, it not only possesses all the convenient formal properties that one expects, but also is effec tively computable.
Basic properties of this equivariant cohomology theory as well as some fundamental general theorems such as the localization theorem of Borel-Atiyah Segal type are formulated and proved in Chapter III.
In Chapter IV, we shall proceed to investigate the relationship between the geometric structures of a given G-space X and the algebraic structures of its equivariant cohomology HJ(X). From the viewpoint of transformation groups, those structures which are usually summarized as the orbit structure are certainly the most important geometric structures of a given G-space X. Hence, it is almost imperative that one should investigate how much of the orbit structure of X can actually be determined from the algebraic structure of HJ (X). Examples of specific problems in this area are: How much of the cohomology structure of the fixed point set, H*(F), is determined by the algebraic structure of Ht(X)? Is it possible to give a criterion for the existence of fixed points purely in terms of HJ(X)? Suppose F=F(G,X)=0 (is empty), how does one determine the set of maximal isotropy subgroups from H~(X)? In the special case of elementary p-groups, we shall formulate various commutative algebraic invariants of HJ(X), which are, then, proved to be intimately related to the orbit structure of the G-space X. No tice that there are general counter-examples for almost all non-p-primary groups which clearly indicate the non-existence of a general relationship between the orbit structure of X and the algebraic structure of Hi; (X). Such a sharp contrast of behaviors between transformations of elementary p-groups and transformations ofnon-p-primary groups is probably one of the most profound as well as fascinating facts in the cohomology theory of transformation groups. In retrospect, this also explains why torus groups play such a central role in the representation theory of compact connected Lie groups, which, after all, is concerned with the special case of linear transformation groups.
Methodologically, one of the central themes in the approach of this book is that the cohomology theory of topological transformation groups can be developed roughly along the same lines as the classical linear representation theory of compact connected Lie groups. A concise exposition of the theory of compact Lie groups and their linear representations is given in Chapter II. In order to present the theory of linear transformation groups as a prototype of cohomology theory of topological transformation groups, we purposely adopt a rather geometric approach, in which the orbit structure of the adjoint action plays the central role. Moreover, it will be clear from such an exposition that the following two basic theorems constitute the foundation of linear representation theory of compact connected Lie groups:
Introduction VII
(i) Structural splitting theorem for linear tori actions: every complex linear representiation of a torus group always splits into the sum of one-dimensional representations.
(ii) Maximal tori theorem of E. Cartan: the set of maximal tori forms a single conjugacy class and G= U{gTg- 1 ; gEG}.
The first result classifies linear tori actions in terms of an extremely simple invariant called the weight system and the second result reduces the classification of linear actions of a compact connected Lie group G to the restricted actions of its maximal tori. Correspondingly, in the setting of the cohomology theory of topological transformation groups, the above structural splitting theorem for linear tori actions can be generalized into various structural splitting theorems of the equivariant cohomology (cf. Chapter IV), which can be considered as the generalized splitting principle in the geometric theory of generalized charateristic classes. Similar to the linear case, one may also combine the structural splitting theorems with the maximal tori theorem to define a (geometric) weight system for topological transformation groups. Such a program is carried out explicitly for the special cases of acyclic manifolds and cohomology spheres in Chapter V; and for cohomology projective spaces in Chapter VI. Although the geometric weight systems are no longer "complete invariants" for topological transformation groups, they nevertheless determine the cohomology aspects of orbit structures of the restricted actions of maximal tori and, hence, also the orbit structure of the original G-action to a great extent.
In Chapter VII, we apply the cohomology method to study transformation groups on compact homogeneous spaces. Due to the fact that compact homo geneous spaces encompass great varieties oftopological types and that the study of transformation groups on them is just getting started, there is an abundance of natural problems in this area, but, so far, only a small number of testing cases have been successfully settled and most of them are as yet unpublished. Therefore, results in this chapter are rather incomplete, and they should be considered only as beginning testing cases that serve to indicate the existence of interesting problems and deep results. In a paper soon to be published, I hope to give a more systematic account of the applications of the cohomology method to the study of trans formation groups on compact homogeneous spaces.
This book is based on a course given at the University of California, Berkeley. I am indebted to the participants of that course, especially to Dr. T. Skjelbred who helped to prepare a preliminary draft of Chapters III and IV.
Berkeley, in Spring 1975 Wu Yi Hsiang
Table of Contents
Chapter I. Generalities on Compact Lie Groups and G-Spaces . 1
§ 1. General Properties of Compact Topological Groups . . . 1 § 2. Generalities of Fibre Bundles and Free G-Spaces . . . . . 6 § 3. The Existence of Slice and its Consequences on General G-Space .... 10 § 4. General Theory of Compact Connected Lie Groups. . . . . . . . . 13
Chapter II. Structural and Classification Theory of Compact Lie Groups and Their Representations. . . . . . . . . 17
§ 1. Orbit Structure of the Adjoint Action. . . . . . 17 § 2. Classification of Compact Connected Lie Groups. 23 § 3. Classification of Irreducible Representations. . . 30
Chapter III. An Equivariant Cohomology Theory Related to Fibre Bundle Theory. . . . . . . . . . . . . . . . . 33
§ 1. The Construction of H~(X) and its Formal Properties. 33 § 2. Localization Theorem of Borel-Atiyah-Segal Type . . 39
Chapter IV. The Orbit Structure of a G-Space X and the Ideal Theoretical Invariants of H~(X). . . . . . . . . . . . . . . . . 43
§ 1. Some Basic Fixed Point Theorems . . . . . . . . . . . . . . 43 § 2. Torsions of Equivariant Cohomology and F-Varieties of G-Spaces 54 § 3. A Splitting Theorem for Poincare Duality Spaces. . . . . . . . 65
Chapter V. The Splitting Principle and the Geometric Weight System of Topological Transformation Groups on Acyclic Cohomology Manifolds or Cohomology Spheres . . . . . . . . . . . . 70
§ 1. The Splitting Principle and the Geometric Weight System for Actions on Acyclic Cohomology Manifolds . . . . . . . . . . . . . . . . . 71
§ 2. Geometric Weight System and Orbit Structure. . . . . . . . . . . 75 § 3. Classification of Principal Orbit Types for Actions of Simple Compact
Lie Groups on Acyclic Cohomology Manifolds. . . . . . . . . . . 83
x Table of Contents
§ 4. Classification of Connected Principal Orbit Types for Actions of (General) Compact Connected Lie Groups on Acyclic Cohomology Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 91
§ 5. A Basis Fixed Point Theorem . . . . . . . . . . . . . . . . . . 95 § 6. Low Dimensional Topological Representations of Compact Connected
Lie Groups. . . . . . . . . . . . . . . . . . . . . 97 § 7. Concluding Remarks Related to Geometric Weight System ..... 101
Chapter VI. The Splitting Theorems and the Geometric Weight System of Topological Transformation Groups on Cohomology Projective Spaces ........................ 105
§ 1. Transformation Groups on Cohomology Complex Projective Spaces . 106 § 2. Transformation Groups on Cohomology Quaternionic Projective
Spaces ............................ 112 § 3. Structure Theorems for Actions of 7lp-Tori on 7lp-Cohomology Pro
jective Spaces (p Odd Primes) . . . . . . . . . . . . . . . . . . 118 § 4. Structure Theorems for Actions of 7l2-Tori on 7l2-Cohomology Pro-
jective Spaces . . . . . . . . . . . . . . . . . . . . . . .. 121
Chapter VII. Transformation Groups on Compact Homogeneous Spaces 129
§ 1. Topological Transformation Groups on Spaces of the Rational Homotopy Type of Product of Odd Spheres. . . . . 134
§ 2. Degree of Symmetry of Compact Homogeneous Spaces 148
References . . 160
Chapter I. Generalities on Compact Lie Groups and G-Spaces
This chapter will briefly review the general facts about compact topological groups, fibre bundles, topological G-spaces and compact Lie groups that are necessary for the subsequent development. Basic concepts and definitions will be adequately explained; and pro<?fs of some fundamental theorems will also be included whenever short clear cut proofs are available.
§ 1. General Properties of Compact Topological Groups
Naturally, a topological group G consists of both a topological structure and ·a group structure which are compatible in the sense that the group structure is continuous with respect to the topological structure. More precisely, the multi plication and inversion mappings are both continuous: G x G-+G, (gl,g2)>-+gl . g2; G-+G, g>-+g-l. Similarly, a Lie group (or rather a differentiable group) G consists of both a differentiable structure and a group structure which are compatible in the sense that the multiplication and inversion mappings are both differentiable. For many problems, the subclass of compact topological groups, or more specifi cally, compact Lie groups plays ·an important role. In this section, we shall sum marize the basic properties of compact topological groups:
(A) Averaging and Haar Measure
Obviously, a finite group G (with discrete topology) is a rather special example of compact topological group. Suppose qJ: G-+G 1(V) is a given representation which represents G as a group oflinear transformations on a vector space V. The following well known "averaging method" is a natural, simple-minded way to show the existence of an invariant inner product on V (with respect to the action of G). Given an arbitrary inner product <x,y) on V, it is clear that the following "averaged inner product" (x,y):
2 Chapter I. Compact Lie Groups and G-Spaces
is an invariant inner product on V with respect to the action of G. And this is also the only effective way, so far, to prove the complete reducibility of representations of finite groups.
Next let us consider the group of unit complex numbers Sl = {ZE<C; Izi = 1}. TopologicalIy, it is a circle and it is one of the simplest example of compact topological group with an infinite number of elements. If we parametrize the circle group in the usual way, i.e. Sl={e21ti8;O~e~1}, and f is a continuous function on Sl, then the integration of f
S6f(e)de
is clearly a generalized "average value of f". Similarly, to any linear representation cp:Sl-'>GI(V) and a given inner product <x,y) on V, the folIowing "(generalized) averaged inner product" is again invariant with respect to the given action of Sl on V.
(x,y) = S6 <e21ti8 • x, e21ti8 • y) de.
As for a general compact group G, it is natural to blend together the above two kind of "averagings". Namely, for a finite subset A={aj}~G and a continuous function f(g), one defines the averaging of f over A to .be
HeuristicalIy, it is reasonable to expect that {A ·f} wilI tend to a constant function, 1(f), as a limit when A becomes more and more dense in G. This is exactly the idea of Von Neumann in defining the (invariant) Haar-integral on a compact group G. We state the result as the folIowing theorem and refer to Pontrjgin's book "Topological Groups" for a detail proof.
Theorem 1.1 (Existence and uniqueness of Haar integral). Let G be a given compact topological group and C(G) be the space of real-valued continuous functions of G. Then there exists a unique continuous linear functional I:C(G)-,>IR. satisfying
(i) (left invariant): l(fa(g))=I(f) for all aEG, where fa (g) = f(ag). (ii) (positive andnormalized): f~O=I(f)~O, and 1(1)=1.
The above invariant linear functional I can also be considered as the integral with respect to the invariant measure-the Haar measure-on G with total volume 1, i.e., 1(f)=SGf(g)dg.
Corollary (1.1.1). Every complex (resp. real) representation of compact group G is equivalent to a unitary (resp. orthogonal) represenation.
Corollary (1.1.2). Every complex (resp. real) representation cp ()[ a compact group G is completely reducible. Hence it decomposes uniquely into the direct sum of irreducible representations.
§ 1. Compact Topological Groups 3
(B) Schur Lemma and Schur Orthogonality Relations
The Schur lemma is simply a neat reformulation of irreducibility purely in terms of operators; however, it is extremely useful and is of fundamental importance.
Schur Lemma. Let Sl' S2 he irreducible subsets of linear transformations 011
Vj. Vz respectively and A: VI -+ V2 he a linear mapping such that A· Sl = Sz . A (set-wise). Then either A =0 or A is invertible.
Proof. It follows easily from A· SI = S2' A that Ker A and Irn(A) are invariant subspaces of S I. S 2 respectively. Hence. the irreducibility of Sl' S2 implies that either KerA=V1• i.e., A=O; or KerA=(O} and Im(A)=V2 . i.e., A is inverti ble. 0
The following special form of Schur's lemma is often useful and deserves special attention.
Special Form of Schur's Lemma. In the special case that SI = S2 (then of course VI = Vz). and the field F is algebraically closed then there exists ;ooEF such that A=)oo·I.
Proof. A· SI = Sl . A=(A -;.1). SI = SI . (A -;.1) for any ;.EF. On the other hand. since F is assumed to be algebraically closed, there exists eigen-value lo E F of A, i.e., (A-}o1) is non-invertible. Hence, by the above lemma (A-)'oI)=O, i.e., A =)'0 1. 0
As a direct consequence, we have the following simple but fundamental theorem for representation of compact commutative groups.
Theorem (1.2). Every irreducihle complex representation cp of a cOl/Jpact commuta tive group G is one-dimensional.
Proof. It follows from the commutativity of G and the irreducihility of cp( G) that for every gEG, cp(g)=A{?J)' I. for some }.(g)E<C.
Thus, cp(G)~ p. I} = the set of scalar multiples. Hence, every subspace is invariant and cp(G) is irreducible only when dim cp = 1. 0
Corollary (1.2.1) (Classification' of irreducible representation of torus group). The set of equiwlence classes of irreducible complex representations of a torus group of rank k, jk ~ IRk /71.\ is naturally in 1 -1 correspondence with the set oj" integral linear functional oj" IRk. Hence. it is also naturally in 1 -1 correspondence with elements oj" Hl(Tk,71.), i.e ..
where i E H l(SI, 71.) is the fundamental class with positil'e orientation.
Definition. With respect to a chosen basis. let a unitary representation cp: G-+ Urn) be given by its matrix form (p(g) = (CPu(g)). Then CPij(g) are called the representation functions of cp.
Example. In the well known classical case of G = Sl, the representation function corresponding to the irreducible representation (p: Sl-+SI with winding number
4 Chapter 1. Compact Lie Groups and G-Spaces
n is e21Cin6, and it is the fundamental fact of Fourier analysis that the set of such representation functions, {e27tin6; n E Z}, of all irreducible representations of Sl forms an orthonormal basis of L2(Sl).
Combining the Schur lemma with the invariant integration, one has the following orthogonality relations for representation functions of a general compact group G.
Theorem (1.3) (Schur orthogonality relations). Let G be a given compact group and cP, ljJ are two non-equivalent, irreducible, complex representations of G. Then
(i) SG CPij(g) ljJkl(g)dg = <CPiig), ljJkl(g) =0,
(ii) SGCPiig)'CPkl(g)dg=<CPij,CPkl) =-d.1 ·bik·bjl · lmcp
Proof. Let V, W be the representation space of ljJ,cP and .2"(V, W);:; V*@W be the space of all linear mappings of V into W. Define the induced representation on .2"(V, W) by
g.A=cp(g).A.ljJ(g)-l=cp(g)A·ljJ(g)t, AE.2"(V, W).
T~en, it is cl~ar that A=SG(g·A)dg is an invariant element, i.e., g.A=cp(g) . A .ljJ(g)-l =A for all gEG. Hence, it follows from Schur lemma that A is either 0 or invertible. But cP,ljJ are non-equivalent, A must be 0 for all AE.2"(V, W). If we write down the equation Ejk=O for the usual basis Ejk of .2"(V, W) in matrix form, we get
Similarly, it follows from the special form of Schur's lemma that
and suitable ABE <C. On the other hand, the invariance of trace under conjugation enables us to compute AB as follows:
dimcp· AB =tr(.8) = SG tr(g B)dg = SG tr(B)dg =tr(B).
Hence, again, we have
(C) Characters: A Complete Invariant of Linear Representations
Definition. X",(g) def tr(cp(g)) = L~= 1 CPii(g) is called the character (function) of the complex representation cP of G.
§ 1. Compact Topological Groups 5
Observations. (i) Since tr(cp(g)) = the sum of eigen-values of cp(g), Xc/g) may be considered as a linear-type invariant of cp(g),
(ii) Xcp+ ",(g) = tr( cp + I/J)(g)) = tr(cp(g)) + tr(l/J(g)) = Xcp(g) + X",(g), (iii) Xcp®",(g) = tr(cp@l/J(g)) = tr(cp(g))· tr(l/J(g)) = Xcp(g)· X",(g), (iv) Xcp(gl ggil)=tr(cp(gl)CP· (g)cp(gl)-I)=tr(cp(g))= Xcp(g)·
One of the most fundamental consequences of Theorem (1.3) is that the charac ter function Xcp(g) completely classifies the representation.
Theorem (1.3'). (i) The characters of two non-equivalent, irreducible complex representations cp and I/J are orthogonal to each other, <XCP' X",) = o.
(ii) A complex representation cp is irreducible if and only if the norm of Xcp is 1, i. e.,
(iii) Two complex representation cp I' CP2 (not necessary irreducible) are equivalent if and only if
Proof. Follows immediately from Theorem (1.3) and complete reducibility.
Remarks. (i) Since the character function Xcp has the same value on conjugate elements, Xcp can be viewed as a function on the set of conjugacy classes GlAd. Furthermore if H c::; G is a closed subgroup of G which intersects every conjugacy class of G, then
(ii) The compactness of G implies that cp(g) are unitary [Cor. (1.1)] and hence diagonalizable. Therefore the equivalence class of cp(g) is completely determined by the basic symmetric forms {oJ of its eigen-values {A J On the other hand, since the set cp(G) is closed under multiplication, the character Xcp(g)=trcp(g) =CT 1 (..1.) not only gives us CT 1 (..1.), but also Xcp(gk)=tr(cp(gt)=L'i=I..1.~, which in turn gives back {CT j ,j=l, ... , n}. Roughly, this is the basic reason why such a simple, linear-type invariant as the character already suffices to classify them. One also sees that both the compactness and the group structure play a vital role.
(D) Completeness Theorem of Peter-Weyl
Theorem (1.4) (Peter-Weyl). For a given compact topological group G, let r be a complete collection of non-equivalent, irreducible complex representation of G. Then the linear combinations of {CPiig), CPEr} are uniformly dense in the space of continuous functions on G, C(G). Or equivalently, {CPiig), CPEr} forms a basis in the Hilbert space of L2-functions on G, L2(G).
We refer the reader to Pontrjgin's book for a standard proof of this theorem.
Definition. A representation cp is called faithful if it is an isomorphism into.
Corollary (1.4.1). Every compact Lie group has a faithful representation.
Proof. Since {CPiig), CPEr} span C(G), they clearly seperate points. Hence, it is a direct consequences of Peter-Weyl theorem that
n {ker(cp); CPEr} = rid} = the identity subgroup.
Therefore n {kercp- U); CPEr} =0
for any neighborhood U of the identity, and it follows from the compactness of G (or rather the compactness of (G- U)), that there exist a finite subcollection CPl' ... , CPk Er with n~=l {kercpi- U} =0.
n~= 1 {kercpJ = ker(cpl + ... + CPk) ~ U .
On the other hand, it is easy to show that for a small neighborhood U in a Lie group, the only subgroup contained in U is the identity subgroup. Hence
1. e., cP 1 + ... + CPk is a faithful representation of G. 0
Remark. (i) Since every closed subgroup of a Lie group is also a Lie group, a compact topological group has a faithful representation when and only when it is a Lie group.
(ii) For a compact group G with a given faithful representation, it is not difficult to deduce the Peter-Weyl theorem of G by using the Stone-Weistrass approximation theorem. Hence, for compact Lie groups, the Peter-Weyl theorem is equivalent to the existence of faithful representations.
§ 2. Generalities of Fibre Bundles and Free G-Spaces
(A) The Concept of G-spaces and Fibre Bundles
Definition. A topological transformation group consists of a topological space X, a topological group G and a transformation of G on X (i. e., a map G x X -> X; (g,x)>--*g' x with e· x=x, gl' (g2' X)=(gl' g2)' x) which is continuous in the sense that the map G x X -> X; (g, x)>--*g· x is continuous. Sometimes, we also call such a structure a topological G-action on X, or simply a topological G-space.
Examples. (i) The linear transformation groups of
GL(n, IR) on IR n, or GL(n, <C) on <cn ,
or their restrictions via some linear representations:
G ..!4 GL(n,IR) (or GL(n, <C))
are obviously topological transformation groups.
§ 2. Fibre Bundles 7
(ii) The isometry group of a Riemannian manifold, (equipped with compact open topology) is a topological transformation group.
Intuitively, a fibre bundle consists of a projection p: B-X from the bundle space B onto the base space X which is locally a product but globally twisted. One of the simplest such example is the Mobius band which is a twisted product of the circle Sl and the interval I with an orientation reversion Z2-twist on I. Technically and theoretically, it is important to pin down precisely what are the permissable twistings. This is exactly where the so called structural group G and the G-space structure on the fibre Y enter into the formal definition of fibre bundles.
Definition. A fibre bundle consists of a bundle space B, a base space X, a fibre Y with a given topological G-action and a projection p: B-X together with compa tible local product structures. Namely, there exists a family of open coverings {V;} of X and local product structures, ¢/V;xY-p-l(V;)~B such that the twistings between two overlabing local product structures are provided by the given G-action on Y, i. e.,
where ¢jl¢;(X,y)=gji(X)'Y' xE(V;nV;), YEY and gji:V;nV;-G is continuous.
Remark. The above definition involves a choice of coordinate neighborhoods {V;} and local product structures {¢;}, hence it is not intrinsic and there is an ob vious kind of equivalence relation among two such structures. We refer to § 2,3 of Steenrod's book [S 11] for a detail discussion on this matter.
(B) Principal Bundle and Principal Map
Definition (of bundle map). Let B,B' be two fibre bundles with the same G-space Y as fibre. Then a continuous map h:B-B' is called a bundle map if
(i) h carries each fibre Yx of B homeomorphically onto a fibre Yx' of B', thus inducing a map Ii: X - X' with p' h = Ii p,
(ii) it is compatible with the twisting in the sense that
v,nhfJlX Y 0"", T,Y p-l(V;nli-l(U)) ~ p'-l(U)
8 Chapter!. Compact Lie Groups and G-Spaces
Remarks. (i) In the simplest case of the bundle Y -> {p t} over a point, the operation of an element gEG, Y->Y:y~g·y, is clearly a bundle map and conversely, every bundle map Y -> Y also concides with the operation of some element of G. Hence, there is a bijection between the set of all bundle maps of Y onto itself and G=GIKo, where Ko={gEG;g·y=yforallYEY} is the ineffective kernel.
(ii) It is obvious that the composition of bundle maps is a bundle map.
Principal bundle. Let B..J4 X be a given fibre bundle with a given G-space Y as typical fibre. Naturally, the set 13 of all bundle maps of the simplest bundle Y -> {pt} into B..J4 X constitutes an important structural invariant. As usual, we shall first try to equip 13 with as many natural structures as possible. Since the composition of two bundle maps is a bundle map, there is a natural (right) action ofG on 13:13 x G-+B, (b,fj)~bog. Moreover, there is a projection p:B->X by assigning each bundle map b to the induced image in the base, i. e., p(b)=pb(Y)EX, and it is not difficult to show that X~BIG (bijection).
Definition (Principal bundle). The principal bundle associated to a bundle B ..J4 X is the set 13 of all bundle maps of Y ->{pt} into B ..J4 X, equipped with the above right G-action and the projection 13 L X ~ BIG.
Remark. It follows easily from the local product structure of B ..J4 X that 13 L X is also locally a product. Hence, it is easy to equip 13 with a unique suitable topology so that the bijection X ~ BIG becomes a homeomorphism.
Definition. The evaluation map 13 x Y L B, (b,y)~b(y) is called the principal map.
Remark. (i) In the case that G acts effectively on Y, i. e., Ko = rid}, then G = G and 13 is a free (right) G-space and the above principal map P:B x Y->B simply identifies B with the quotient space BxGY=Bx Y/,,~", where (b,y)~(b.y-\g·y). In general 13 is a free (right) G-space which can also be considered as a (right) G-space naturally. Then again, the map P identifies 13 x G Y ~ B.
(ii) The principal map P indicates how to recover the bundle B..J4 X from the principal bundle 13 and the given G-space Y. Hence, for a fixed G-space Y, the principal bundle 13 constitutes a complete invariant.
(C) Intrinsic Definition of Bundles and Gleason Lemma
(XxG) has an obvious (right) G-action given by (X,g)·gl=(X,g·gll. A free (right) G-space E is called locally trivial if to every point xEEIG (orbit space of E) there exist a neighborhood U, such that p-l(U) is isomorphic to the obvious (right) G-space U x G. In view of the principal bundle and principal map, it is natural to give the following intrinsic definition of fibre bundle:
Definition. A fibre bundle consists of a locally trivial free (right) G-space 13 (called the principal bundle) and a (left) G-space Y (called the fibre). Then the orbit space X=BIG is called the base space; the quotient space B=BxGY=Bx Y/,,~", where (b, y) ~ (b g- 1, g y) is called the bundle space (or total space) and the following map p is called the projection:
§ 2. Fibre Bundles 9
j- j, B=BxGY ~ B/G=X
where P1 is the projection onto the first factor and vertical map are projections onto respective "orbit spaces".
In many useful cases, the structural group G is either automatically a compact Lie group or can be reduced to a compact Lie group (cf. (E)). In those cases, the following lemma of Gleason adds a convenient bonus to the above definition, namely, the local triviality condition holds automatically for any free G-space when G is a compact Lie group.
Gleason Lemma. ~f G is a compact Lie group, then any free G-space is locally trivial.
(We shall prove a more general form of this lemma in § 3.)
(D) Homotopy Lifting Property, Induced Bundles and Classifying Bundles
For the purpose of homotopical investigations, the following direct consequence of the local product structure of fibre bundles is of fundamental importance.
Homotopy lifting property. Let B ~ X be a fibre bundle and K be a finite polyhedron. Let fa: K -> B be a map and 1: K x I -> X be a homotopy of p -fa = la. Then there exists a homotopy f: K x 1-> B covering l (i. e., p f =f), and f is stationary with .r.
The proof of the above property is simple and straightforward by using the local product structure of p: B -> X. However, it is so fundamental that it led 1. P. Serre to axiomatize it in defining fibre spaces [S 3J.
Induced bundle. Let B ~ X be a fibre bundle and f: X' ->X be a map. Let B' be those points (b,x') of B x X' with p(b)=f(x') and l,p' be the restriction of projections to B', i. e.,
Then, it is easy to check that B' ~ X' is a fibre bundle and .1 is a bundle map. B' ~ X' is called the induced bundle of B ~ X with respect to .f.
Classification Theorem. Let E'G ~ B'G be a principal G-bundle with n,(E'G) = 0 for 0 ~ i ~ n, and K be a finite polyhedron of dim ~ n. Then the operation of assign ing to each map f: K -+ B'G its induced bundle sets up a bijection between homotopy classes of maps of K into B'G and equivalence classes of principal G-bundles over K.
We refer to § 19 of [S 11] and [M 2] for a proof of the above theorem and the general existence of such classifying (universal) bundle.
(E) Reduction and Extension of Structural Group
Let H s;; G be a closed subgroup of G and B ~ X and B' L... X be respectively H-bundle and G-bundle over X (i.e., free H-space and G-space with X as their common orbit space). If there exists an equivariant map r:B-B' which commutes with projections, i. e.,
B~B'
\} X
commutes and r(b· h) = r(b)· h, then B ~ X is called a reduction of B' L... X and B' L... X is called an extension of B ~ X.
Observe that B' =B' xGG and hence B'jH =B' xG GjH. Therefore, a reduction r:B->B' induces the following maps
BjH ~ B'jH=B'xGGjH
\! X
which gives a cross-section r·p-l:X-B'x GGjH. Conversely, suppose s:X -+B' xG GjH =B'jH is a given cross-section. Then the inverse image of s(X) of the projection Po:B'-+B'/H, i.e., B= P0 1(S(X)), is clearly a free H-space with s(X)=X as its orbit space and the inclusion B=P01(S(X))S;;B' is equivariant. Hence,
Proposition. For a given closed subgroup H S;; G, there is a 1 -1 correspondence between the reductions of a G-bundle B' -> X to an H -bundle and the cross-sections of the associated GjH-bundle B'jH =B' xG GjH -X.
§ 3. The Existence of Slice and its Consequences on General G-Space
From now on, we shall always assume that G is a compact Lie group. The following simple, useful fact was first noticed by Koszul [K 3] and then further generalized and exploited by Montgomery-Yang, Mostow and Palais. [M 4, M 9, P 1]
§ 3. Slices and G-Spaces 11
(A) Differentiable Slice
Theorem (1.5). Let M be a differentiable G-space, H = Gx be the isotropy subgroup at a point xEM, and ({Jx be the local representation oj H on normal vectors (w.r.t. a chosen invariant metric) oj the orbit G(x)~ G/H at x. Then, the equivariant normal bundle v(G(x)) oj G(x) is isomorphic to
where H acts on G as right translations and on lRk via ({Jr Furthermore, Jor a sufficient small e>O, the exponential map (w.r.t. a chosen invariant metric) is an equivariant diffeomorphism oj the associated e-disc bundle oj a(({Jx): G xHD:--.G/H onto an invariant tubular neighborhood oj G(x).
ProoJ. Let lR~ be the set of normal vectors to G(x) at x (w.r.t. a chosen invariant metric, which always exists by means of averaging). Then the induced G-action carries lR~ to lR~.x' It is easy to check that gt'Vl=gZ'VZ if and only if (gt,v 1)
and (gz, Vz)E G x lR~ are equivalent in the usual sense, i. e.,
~ G x T(M) ---> T(M)
Hence, the above theorem follows from a straightforward verification. 0
(B) Gleason Lemma and the Existence oj Topological Slice
Lemma (3.1). IJ H is a closed subgroup oj G (a compact Lie group), then there exists a finite dimensional linear G-action with a vector v such that H = Gv is the isotropy subgroup oj v.
Proof. It follows directly from Peter-Weyl theorem and the descending chain condition for closed subgroups of a compact Lie group. We refer to the paper of Mostow [M 9J and Palais [P 1 J for a detail proof. 0
Gleason Lemma (Generalized Form). IJ K is a compact invariant subspace oj a G-space X and if J: K --. V is an equivariant map oj K into a linear G-space, then J admits an equivariant extension j: X --. V.
ProoJ. By Tietze's extension theorem, there is a continuous extension J*: X --. V. Let J(x)= JGg- 1 J*(gx)dg. It is easy to check that J is an equivariant extension of f.
Theorem (IS) (Existence of a slice). Let X be a topological G-space and H = Gx'
Then there exists a subset S oj X such that (i) S is invariant under H, (ii) G(S) = G x H S and is an invariant neighborhood oj G(x) in X. Such a subset is called a slice at x.
Proof. In the special case that X is a differentiable G-space, a small normal disc D~(e) is clearly such a slice at x. In the general topological case, we first equi variantly embed the orbit G(x) ~ G/H into a suitable linear G-space V,f: G(x) ~G/H ~ G(v)s; V (by lemma (3.1)). Then extend f to an equivariant map j: X --+ V. Let S' be a slice at v (which exists because V is linear) and put S = J -I (S'). Then it is easy to check that S is such a slice. 0
The above slice theorem has the following two consequences of basic im portance:
(C) Equivariant Embedding
Theorem (1.6) (Mostow-Palais). If G is a compact Lie group and X is a separable, metrizable G-space of finite dimension and with finite orbit types, then X admits an equivariant imbedding into a linear G-space.
Since we actually do not need the above theorem in any essential way, we refer to [M 9J or Ch. VIII of [B 10J for a proof.
(D) Principal Orbit Type Theorem
For a given G-space X, the set of all isotropy subgroups (D(X)={Gx;XEX} clearly divides into conjugacy classes which correspond to the types of orbits in X. Observe that a homogeneous space G/K can map equivariantly onto G/H if and only if K is conjugate to a subgroup of H, i. e., gKg- I s;H for suitable gEG. Hence, it is natural to introduce the following partial ordering relation among the set of orbit types:
(D(X)= {Gx: XEX} = U(H;) (conjugacy classes), (Hi)?: (Hj)~;""Hi is conjugate to a subgroup of Hj.
Theorem (1.7) (Montgomery-Samelson-Yang). Let M be a connected manifold with a given differentiable G-action. Then, among the set of orbit types (D(M) = U(H;), there is a unique maximal orbit type (HI) such that
(i) (HI)?:(GJ for all XEM, (ii) the union of all orbits of (H1)-type=M(Hd={x;GxE(H I)) is open dense
in M and the codimension of (M -M(Hd) is at least 1. (iii) F(HI,M) intersects every orbit, (iv) the orbit space M(HjG is connected.
Proof. Let M' =M/G be the orbit space and (D(M) be the set of orbit types (with discrete topology and partial ordering). Let us conSider the orbit type function Q:M'--+(D(M). It follows from the differentiable slice theorem that x'=G(x) is a local maximal if and only if ({Jx is a trivial representation (i. e., Gx acts trivially on the slice Sx = D~, and hence Q is locally constant in a neighborhood of those local maximal points x'. On the other hand, suppose y' = G(y) is not a local maximal, then there are the following two cases:
§ 4. Compact Connected Lie Groups 13
(1) codim(F(G)',S))~2, then Sy-F(Gy,Sy) is still connected and hence if we remove those orbits of the same type as y' from the neighborhood of y', S)Gy, the remaining set [Sy - F( Gv, S.vlJ/Gy is still connected.
(2) codimF(G)"Sy)=1, then qJy:Gy ...... O(1)~~2 and acts on Sy as a reflection with respect to the hyperplane F(Gy,S). Hence the orbit space S)Gy is a half plane with the image of F( Gy, Sy) as boundary. Therefore [Sy - F( Gv, SyJ/G,. is again connected.
Since M is assumed to be connected, M' = M /G must be also connected. The above fact shows that the removing of all those points y' which are not local maximals did not even separate the space M' locally. Hence, the subspace M~ of all local maximal points x' of M' is still connected. But Q is locally constant on M~, Q must be in fact a constant on M~. That is, there is a unique (local) maximal orbit type (Hi) and M~ = M(HJl/G is connected, open dense in M'. The assertions (ii), (iii) also follow immediately. 0
Remark. (i) The above theorem also holds without modification for topological G-action on cohomology manifold over ~. It also holds for connected orbit types, (i. e., the types of connected isotropy subgroups G~) for topological G-action on cohomology manifold over <Q. Such generalizations will be proved when we are ready to show the same fact about codim F(Gy,S) by cohomology method.
(ii) The unique maximal orbit type is called the principal orbit type, and its corresponding isotropy subgroups are called principal isotropy subgroups.
§ 4. General Theory of Compact Connected Lie Groups
(A) Adjoint Action and Adjoint Representation A one-parameter subgroup of a Lie group G is a (differentiable) homomorphism, ~, of the simplest Lie group JR 1 into G. Clearly, the right translations of {~(t), tE JR l} exhibit a left-invariant (due to the associativity of G) JR I-action on G. Hence, the velocity vector at every point forms a left-invariant vector field X on G. Conversely, if we integrate a left invariant vector field X, then it is easy to show (by uniqueness and left invariance of integral curves) that the integral curve passing through identity is a one-parameter subgroup. Hence, there are the following bijections
{one-parameter subgroups} +-> {left invariant JRi-actions} t t
{tangent vectors at identity} +-> (left invariant vector fields} .
We identify them via the above bijections and call it the Lie algebra, g, of G, which is a vector space with a bilinear bracket product [X, YJ (of vector fields) satisfying the following Jacobi identity:
[[X, YJ,ZJ + [[Y,ZJ,XJ + [[Z,XJ, YJ =0.
Furthermore, It IS convenient and useful to organize all one-parameter sub groups into a (universal) map:
Exp:g-+G
such that Exp(t X): IRl-+g-+G gives the one-parameter subgroup with X as its velocity vector at the identity e.
Remark. (i) Recall that the geometric meaning of bracket product [X, Y] of two vector fields X, Y is the following: If one "drifts" successively in the flow X, Y,( - X) and then (- Y) each for a time t, one ends up travelling approx imately t2. [X, Y]. Hence, in our case of left-invariant vector fields, we have [X, Y]e = the velocity vector (at e) of the curve
y(t) = Exp Vi X. Exp Vi y. Exp( - Vi X)· Exp( - Vi Y).
t2-[X. Y] (I)
(ii) If h: G1 -+G2 is a homomorphism of Lie groups, then the differential of h at e clearly induces a homomorphism of their Lie algebra dhe :g1 -+g2 ,
Definition. The conjugations represent G as a differentiable transformation on itself via inner automorphisms, i.e., GxG-+G, (gl,g2)>-+glg2g1 1• We shall call it the adjoint G-action on G itself. The above adjoint action on G induces another adjoint action on its Lie algebra g, which is a linear representation of G into GL(g), Ad: G-+GL(g) (general linear group of g), called the adjoint representa tion of G. Furthermore, the differential of Ad at e, ad = Ade : 9 -+g /(g) is called the adjoint representation of g. In terms of exponential maps, the above two adjoint representations of G and 9 are respectively characterized by the follow ing identities
(i) Expt· [Ad(ExpsX)· Y] =ExpsX· Expt Y·Exp( -sX), (ii) Ad(ExptX)· Y = Exp(t·ad(X)). Y.
Proposition (4.1). ad(X)· Y = [X, Y] for all X, Y Eg.
Proof. It follows from the geometric interpretation of the bracket product that
ExpsX ExptY·Exp( -sX)·Exp( -tY) = Expst· [X, Y]+O(st).
§ 4. Compact Connected Lie Groups
On the other hand, by the above definition, we have
Exp[(Exps ad(X) - I)· t Y] = Exp(st· ad(X)· Y)+O(s t)
II Exp(t Exps ad(X)· Y)· Exp( - t Y)+O(st)
II by (ii) by (i)
Expt[Ad(ExpsX). Y] Exp( -t Y) = ExpsX· ExptY· Exp( -sX)Exp( -tY)
= Expst·[X, Y]+O(st).
Hence, if we take the limit s,t~O, we get ad(X)· Y = [X, Y]. 0
(B) Cart an's Theorem of Maximal Tori and its Consequences
15
Theorem (1.8) CEo Cartan). Let G be a compact connected Lie group and T be a given maximal torus. Then
(i) all maximal tori of G are mutually conjugate and they are exactly those connected principal isotropy subgroups of the adjoint action on G.
(ii) T intersects with every conjugacy class (i. e., orbit of adjoint action) or equivalently G=U{gTg-:-1;gEG}.
Proof. The above theorem is a direct consequence of Theorem (1.7) and the fact that maximal tori are exactly those connected principal isotropy subgroups of the adjoint action on G. Since principal orbits are everywhere dense and the adjoint action on 9 is precisely the local linear action around the identity eE G, it is equivalent but technically simpler to prove that the maximal tori are the connected principal isotropy subgroups of the adjoint action on g. The follow ing lemma provides an effective way of computing the principal isotropy sub groups of a linear action.
Lemma (4.1). Let ljJ be a linear G-action and ({Jx be the slice representation of Gx at x (cf. § 3-A). Then
Proof. It is clear that the local representation of Gx at origin is ljJ I Gx ' while that of x is ({Jx (:normal part)+(AdGIGx-AdG )(:tangent part). But the fixed point set F(Gx ' V) is a linear subspace and hence obviously connected. Therefore the local representations of Gx at origin and at x must be equal. 0
Let X be an arbitrary element of g, Gf be the connected isotropy subgroup of X and gx be the Lie algebra of Gx . It is easy to show the existence of at least a maximal torus T;2 Exp t X, hence Gx contains at least one maximal torus for any X E g. On the other hand, it follows from the above lemma that Gf is a connected principal isotropy subgroup if and only if
is a trivial representation which (in view of the fact Gx contains at least a maxi mal torus), is equivalent to saying that Gf itself is a maximal torus. 0
Corollary (1.8.1). Exp: g --+ G is onto for a compact connected Lie group.
Proof. Let gEG be an arbitrary element then there exists gl EG such that glgg11ET or gEg1 1 Tg1 and hence, the above corollary follows from the obvious fact that Exponential map, is onto for a torus.
Corollary (1.8.2). Gx (w. r. t. the adjoint action on g) is connected for every X Eg.
Proof. Observe that Gx = centralizor of the torus subgroup T(X) generated by {ExptX}. We claim that Gx = the union of all maximal tori containing T(X) and hence connected. Let gEGx be an arbitrary element of Gx . Then g, T(X) generates a compact abelian subgroup which is, in general, a cyclic group prod uet with a torus, i.e., <g, T(X) =Zh x T1. It is easy to show that Zh x Tl =<gl) is a topological cyclic group. Hence, by the above theorem, there exists a maximal torus T2<gl)=<g,T(X), i.e., gET2T(X). 0
Corollary (1.8.3). Let Tr;;;,G be a maximal torus of a compact connected Lie group G. Then two representations q>, IjJ of G are equivalent if and only if their restrictions to T, q> I T and IjJ I T are equivalent.
Proof. q>-1jJ <::> X",=X", <::> X",IT=X",IT=X"'IT=X",IT<::> q>IT-IjJIT (for character functions take constant value on each conjugacy class and T intersects every conjugacy class). 0
Definition. The dimension of maximal tori of a compact Lie group G is called the rank of G.
(C) Root System and Weight System
Definition. For a given complex representation IjJ of a compact connected Lie group G and a fixed maximal torus T, it follows from the above corollary and Schur lemma that IjJ I T is a complete invariant and
. l/t I T = IjJ 1 ff)1jJ 2 ff) .. . ff)1jJ n
splits uniquely into the sum of one-dimensional representations. Hence, the collection of integral linear functionals wjEHl(T,Z) corresponding to IjJj forms a complete invariant of 1jJ, called the weight system of 1jJ, and denoted by Q(IjJ).
Definition. The (non-zero) weight system of the complexification of the adjoint representation of a compact connected Lie group G is called the root system of G, and is denoted by LI(G), i.e.,
LI(G)=Q(AdG®<C)-{those zero weight vectors}.
Remark. The exclusion of zero weights in LI(G) is purely for notational con venience.
Chapter II. Structural and Classification Theory of Compact Lie Groups and Their Representations
This chapter consists of a concise exposition of the structure and classification theories of compact Lie groups and their representations from the geometric viewpoint of transformation groups. An explicit and neat understanding of the orbit structure of the adjoint action of a compact Lie group G plays a central role in the classification theory developed by E. Cartan and H. Weyl. This more geometric approach is actually more natural and straightforward than the usual Lie-algebra-theoretical approach. Furthermore, such an approach will also provide us with valuable examples and insight for later investigation of top ological transformation groups.
§ 1. Orbit Structure of the Adjoint Action
(A) The simplest fundamental examples: SU(2) or SO(3)
The special unitary group of rank 2, SU (2) consists of those 2 x 2 matrices
u = G -;) with det(u) = lal 2 + Ibl 2 = 1. It can also be identified with the group
of unit quaternions, Sp(1)={q=a+j-b,lqI2=lalz+lblz=1}, which is geo metrically the unit sphere S3 (1) s; IR 4. The center of Sp( 1) consists of {± 1 } and the adjoint representation of Sp(l) maps Sp(t)/{ ± l} isomorphically onto SO(3), the rotation group of euclidean 3-space. Geometrically, the adjoint action on Sp(1)=S3(1) is simply the SO(3)-rotation with the real axis as the fixed axis.' Since the total volume of the unit 3-sphere is 2nz, the normalized Haar measure on Sp(l) is the usual measure on S3(1) modified by a constant 1/2n 2 •
Now let us apply the above understanding of the geometric structure of Spit) = SU (2) to the classification of irreducible complex representations of Sp(l).
(i) Let 1j;1: Sp(t)~SU(2) be the usual complex linear action of SU(2) on ([z = {(ZI , Z 2)}' Then the above linear substitutions induce a complex linear action, Ij;k' of Sp(1)~SU(2) on the space of degree k homogeneous polynomials PkS;([[ZI'ZZ] for each k;;;,O.
(ii) Let Tl={a=e2ni8} = unit complexess;Sp(l) (unit quaternions). Then
Tl is a maximal torus ofSp(l) and Ij;I(a)=(~ ~~1} i.e.,
Ij;t(a),zj =a·z1 , Ij;I(a)'zz=a~l·z2'
18 Chapter II. Classification Theory of Compact Lie Groups
Therefore, for the basis {z~, ... , z~ -[ z~, ... , z~} of Pk ,
Theorem (11.1). {'fk' k;?: O} constitutes a complete collection of (non-equivalent) irreducible representations of Sp(l) = SU(2).
Proof. Let d(J be the usual measure on SJ(1) and f be a function of G = Sp(1) constant on every conjugacy class (i. e., a central function). Then, we have the following integration formula
SGf(g)dg = J, Ss3f(g)d(J = ~ S612 f(e 21ti O).4n sin2(2ne)·2nde 2n- 2n
= ~ J6f(e21tiO)'le21tie - e- 2rri012. de = ~ ST'f(t)'IQ(tW dt
where "dg" and "dt" are the normalized Haar measure of G=Sp(l) and TI respectively and Q(t)=(t-t) for t=e21ti OE TI.
(i) Applying the above formula to f(g) = Xk(g)· Xk(g), one gets
SG Xk' Xk dg = ~ SrI Xk(t)· Xk(t)· Q(t)· Q(t) dt
= ~ SrIIXk(t)· Q(tW dt
= ~ Srlltk+ 1 _ t-(k+ 1)1 2 dt = 1.
Hence, it follows from Theorem (1.3') that I/Ik are irreducible for all k;?:O. (ii) Notice that diml/lk=(k+ 1) are different for different k, it is obvious
that {l/Ik,k;?:O} form a non-equivalent family. On the other hand, it is a well known fact in Fourier series that
forms a basis of the sub-space of odd functions in L2 (T 1). Hence, {l/Ik,k;?:O} is already a complete family in the sense that any irreducible complex representa tion, 1/1, of Sp(1) must be equivalent to one of I/Ik' For otherwise. it follows from Theorem (1.3') that
<X'fr' Xk) L2(G) = SG XI/I' Xk dg
=~Srx"f(t)Q(t)'Xk(t)'Q(t)dt=O for all k.
§ 1. Orbit Structure of Adjoint Action 19
Since XIjJ(t)· Q(t) is a continuous odd-function, the above orthogonal relations for the whole basis {Xk(t)· Q(t)} imply that XIjJ(t)· Q(t) == 0, which is obviously a contra diction, for XI//(e) = dim IjJ # O. 0
Remark. (i) The weight system of IPk form a string leading from the highest weight vector k D to - k D, i. e.,
Q(ljJk) = {kO,(k-2)D, ... , (k-2f)D, ... , -kD}.
(ii) IjJk factor through SO(3), i. e., Ker(ljJk) = {± 1}, if and only if k is even. And they forms a complete family of irreducible representations of SO(3).
(iii) In either case of Sp(1), or SOC}), the weight system of any representation consists of at least one non-negative weight which is less than the positive root, i.e., O~w<ct.
Corollary (H.i.l). Every non-commutative compact connected Lie group of rank one is either isomorphic to Sp(1) or to SO(3).
Proof. Observe that the Lie algebras of Sp(1) and SO(3) are isomorphic and can be expressed in terms of basis 91=(H,X,Y) with [H,X]=Y, [H,Y]=-X. and [X, Y]=H. Now, let G be a non-commutative compact connected Lie group of rank one and 9 be its Lie algebra. Let Tl S G be an arbitrary, fixed maximal torus of G, and AdlT I be the restriction of the adjoint representation to Tl. Then
AdITI=1+LJ=1(Xj and 9=lRl +LJ=llR;j
where lR I is the Lie algebra of Tl and (Xj: T I ---+SO(2) is a homomorphism with winding number nj>O; nl~"'~ns' Let 9~=lRl+lR;, and H' the integral base vector of lR I, i. e., 7l. H' = Ker{lR I exp ) T I), and X', Y' be an orthogonal basis of lR;,. Then, by definition,
(E H') (cos2nnl t (Xl xpt· = .
. sm2nnl t -sin 2n 111 t) .
cos2nnl t
Hence, by Proposition (4.1) of § 1.4-A, we have
[H',X'] =nl Y' and [H', Y'] = -nl X'.
The fact that G is of rank one implies that [A, B] = 0 if and only if A, Bare linearly dependent in 9. Hence [X', yI] #0, and
[H', [X', Y']] = - [X', [Y',H']] - [Y', [H',X']]
= - [X',n l X'] - [Y', nl Y'] = 0
imply that [X', Y'] = A' H' for a suitable ;e # O. Then, it is easy to modify the basis {H',X', Y'} by suitable constant to obtain a basis {HI,XI, Yl} of 9~ so that [HI ,Xl] = Yl , [HI' Yl ] = -Xl' [Xl' Yl] =Hl · Hence 9~ ~ 91 as Lie algebra and
consequently, the subgroup G~ with g; as its Lie algebra is isomorphic to either Sp(1) or SO(3). We claim that G~ = G. For otherwise, it follows by applying the above theorem to the isotropy representation of G~ on tangent vectors of GIG; that its weight system Q={cxz, ... ,cxJ must contain at least one 0:s;cxj<!X1
(cf. Remark (iii)) which is a contradiction to the choice that !X! :S;!Xj for all 2:S;j:S;s. 0
(B) An Elementary Structural Theorem for Groups Generated by Reflections
Let M be a connected differentiable manifold. A diffeomorphism I' of M onto itself is called a reflection if 1'2 = identity, the fixed point set of r, F(r) is of co dimension one and (M - F(r)) consists of two connected components inter changed by r.
Theorem (11.2). Let r be a finite group generated by reflections and ~ = {rj }
be the set of all reflections in r. Then r acts simply transitively on the connected components of (M - U {F (r); rj E f!lt}) (called chambers). The closure of an ar bitrary . chamber Co is a fundamental domain of r in the sense that every point xEM is conjugate to exactly one point Xo=Y'XECo '
Proof. Observe that y·F(r)=F(y·r·i)-l) and hence U{F(r),riEf!lt} is invariant under r, and consequently (M - U{F(r), IjE..:.qf}) is also invariant. Suppose Ci
and Cj are two chambers with (C;r\ F(r) n C) of co dimension one. Then it is obvious that r(CJ=Cj. Since the deletion of all intersections of different hyper planes {F(r;) n F(r), ri "" rjE f!lt}, (which is of codimension :s; 2), fails to disconnect M, it is not difficult to show that r acts transivitely on the set of all chambers, i. e., r·Co=(M-U{F(r),rjE.g)!}). Hence r·Co=(M-U{F(r)})=M. We refer to [H 8J for a detail proof that Co intersects every r-orbit exactly once, i. e., Co is a fundamental domain. 0
Corollary (11.2.1). Let ~o be the set of those reflections in r with their hyperplanes F(r) have codimension one intersection with Co. Then r is already generated by those reflections {rEf!lt o}, called a simple system of generators of r.
(C) Weyl Group and a Fundamental Reduction
Definition. Since all maximal tori of a compact connected Lie group G are con jugate, we may arbitrarily choose a maximal torus T~ G and define the Weyl group of G, W(G) =N(T)/T, where N(T) is the normalizor of T in G. Clearly, the conjugations induce an action of W(G) on T and consequently also on its Lie algebra 1), the Cartan subalgebra. The importance of the Weyl group lies in the following fundamental reduction.
Theorem (11.3). Let G be a compact connected Lie group, T a maximal torus of G, L1 (G) the root system of G, and W the Weyl group of G. Then
(i) the multiplicity of each root cx E L1 (G) is one and k· cx E L1 (G) !ff k = ± 1, (ii) the action of W on 1) (resp. on T) is a group generated by reflections
{ra' CXEL1(G)}, where r, is the reflection w.r.t. the hyperplane H,=Ker(1) ~ JR.!) (resp. Ta = Ker(T ~ U(l ))),
§ 1. Orbit Structure of Adjoint Action 21
(iii) Wx = W(GJ and the following inclusions induce bijections on their respective orbit spaces, i. e.,
i~j l)jW ~ gjAd
j j TjW ~ GjAd
Proof. (i) Let Y,,=KerO{T ~ U(l)) and Ga=NO(TJ, the connected normalizor of Ta.' G = GJY". Then it is clear that Ga is of rank one where root system L1 (Ga)
consists of precisely all those roots in L1 (G) proportionate to rt.. However, it follows from Corollary (11.1.1) of §1-A that L1(Ga)={±rt.}, hence the multiplicity ofrt. must be 1 and krt.EL1(G) iff k = ± 1.
(ii) Clearly, F(Ta,g)=ga (the Lie algebra of Ga=N°(TJ=Z(Y,,)) and the induced action of Ga on ga is the adjoint action, AdG ' which is, effectively the rotation of Ga on ga with Ha (co dim 3 in ga' codim 1 in l») as fixed point set. Hence, it is not difficult to see that F(Ga, g) =F{ra,F(T, g)) =F(ra, l») = Ha, where ra is the reflection that generates W(Ga)~Z2.
(iii) Consider the Weyl group W as an (effective) transformation group on l) =F(T, g), W contains the above reflections ra, for each pair of roots ±rt.EL1(G). We claim that W is, in fact, generated by the above reflections {ra}. In order to show that, let W' be the subgroup generated by {ra, ±rt.EL1(G)}. Since the set L1(G) is invariant under W, it is clear that (l)-U{Ha}) is also invariant under W, consequently, W permutes its connected components, i.e., the (Weyl) chambers. Notice that W' permutes the set of chambers simply transitively. Hence, we need only to show that W also permutes the set of chambers simply transitively (for then, ord W=ord W'=the number of chambers). Suppose the contrary. Then there exists an element aE W of order p (prime) such that a(Co)= Co. Hence, it follows from a theorem of P. A. Smith (and the acyclicity of Co) that there exists X E Co fixed under a and consequently, Gx is disconnected, (for G~ = T, GxjG~ 2{a}) which is a contradiction to Corollary (1.8.2) of§ I.4-B. Hence W= W' and is generated by the reflections {ra}.
(iv) It follows directly from the principal orbit theorem that the fixed point set of a principal isotropy subgroup, F(T, g) = l) (resp. F(T, G) = T) intersects every orbit. Namely, the following inclusions induces surjections of the respective orbit spaces:
l)
I GjAd
On the other hand, the injectivity of the above maps simply means that F{T, G(X))=N(T, G)jN(T, Gx )= W(G)jW(Gx) which is a direct consequence of
maximal tori theorem. Finally, we remark that
Definition. The orbit space g/ Ad ~ l)/W is called the Weyl chamber of 9 and the orbit space G/Ad~ T/W is called the Cm·tan polyhedron. In view of Theorem (II.2), it is also customary to identify the above orbit spaces with anyone of their respective fundamental domains.
(D) The Volume Function and Weyl Integration Formula
If one equips G with a bi-invariant Riemannian metric with total volume 1, and lis a central function on G, then the Haar integral SG/(g)dg can be reduced to the following weighted integration
Sp/(t)p(t)dt =! Sr/(t)p(t)dt w
(w=ord(W(G)), and T consists ofwP's)
over the Cartan polyhedron P with p(t)=q-dim volume of G(t), q=dimG/T. Now let us compute the above volume function pet): T -+R as follows. Let us equip G/T with the homogeneous Riemannian metric such that the metric on the tangent space at the base point is the restriction of metric on 9 to l)-L. For a given tE T, we have
UI UI
and it is clear that p(t)=vol(G(t))=vol(G/T) ·IThe Jacobian of let) at el. Since the tangent space at the base point eE G/T, l)\ decomposes into invariant root spaces W.r.t. dl(t)le:
one has det(dl(t))e = TIml+ det(dl~(t))e' In view of the computation for the rank one case, it is not difficult to see that det (d I,U)) = constant· sin 2 (n aCt)) and hence
p(t)=co' det(dl(t))e=C' TIml+ sin2 (n· a(t))=c'Q(t)· Q(t)
Remark. It is useful to note that Q(t) is antisymmetric w.r.t. W, I.e., Q(O"(t)) =sign(O")·Q(t). Then it is easy to show
§ 2. Classification of Compact Lie Groups 23
where c5 =t L'E1 + r:t.. Use the above expression of Q(t). it is easy to determine the constant c' as follows
S 1 S '2 c' 2) 1 = G 1, dg = - T c 'I'Q(t)1 dt = - (IQ(t)IL2 (T) W W
Hence, we have the following important formula,
Weyl Integration Formula. For a central function f on G, one has
SGf(g)dg = ~ h.f(t)IQ(tWdt w
(A) Cartan-Killing Form and Characterization of Compact Lie Algebras
Definition. A (real) Lie algebra g is called a compact Lie algebra if it is the Lie algebra of a compact Lie group G,
Obviously, a compact Lie algebra, g, has an inner product invariant under the adjoint action of G. Namely,
(Ad(ExptX)·Y,Ad(ExptX)·Z)=(Y,Z), forall X,Y,ZEg.
The above identity is clearly equivalent to its differentiated version;
d . - (Ad(Exp t Xl' Y, Ad(Exp t X), Z), 0 =([X, Y],Z) +(y, [X,Z]) =0, dt
An important, simple consequence of the above fact is the following:
Theorem (UA). A compact Lie algebra 9 decomposes uniquely into the sum of its center go and normal simple subalgebras: 9 = go + L gj'
Proof. It follows easily from the above identity that the perpendicular space f.L of an arbitrary normal subalgebra f is also a normal subalgebra, and 9 = f + f.L (as Lie algebras), for XEg, YEt, ZEf.L
[X, Y] d=>(Y, [X,Z])= -([X, Y],Z)=O=[X,Z] Ef.L. 0
In view of the above theorem, we shall from now on assume that 9 is itself simple, although most of the following discussions are also valid (with some
obvious modifications) for general compact Lie algebras, or at least for semi simple compact Lie algebras.
Since a linear subspace f of a Lie algebra 9 is invariant (w.r.t.Ad) if and only iff is a normal subalgebra, the adjoint action of G on its Lie algebra 9 is irreducible if and only if G (resp. g) is simple. Hence, for a simple compact Lie algebra g, two invariant symmetric bilinear forms on 9 are proportional, i. e., Bl (X, Y) =kB2 (X, Y). On the other hand, the following Cartan-Killing form
B(X, Y) def Tr(adx · ad y )
is clearly an intrinsically defined, symmetric bilinear form, and consequently, also invariant (w.r.t. inner automorphisms of g). Hence, the above Cartan-Killing form B(X, Y) is proportional to an invariant inner product on g. Note that adx is anti-symmetric and the eigenvalues of adx are imaginary, therefore the eigen values of (adx)2 are negative, and B(X, X) =tr(adx)2 <0. Hence, B(X, Y) is negative definite, and it is natural to consider (X, Y) = - B(X, Y) as the intrinsic inner product on g. In fact,it is not difficult to prove the following characterization of compact Lie algebras:
Theorem (11.5). A simple (or semi-simple) real Lie algebra 9 is compact if and only if its Cartan-Killing form B(X, Y) is negative definite.
(B) System of Simple Roots and Dynkin Diagram
Let 9 be a simple compact Lie algebra, 1) be an arbitrarily chosen but fixed Cartan subalgebra, and LI s;;1)* (the dual space of 1)) be the root system of g. Let (X, Y) = - B(X, Y) = - tr adx . adx be the intrinsic inner product on 9 and respectively the induced inner products on 1) and 1)*. Then the Weyl group W acts on 1) (resp. on 1)*) as an orthogonal transformation group generated by the reflections {ra, ±IXELI}. Note that since W permutes the chambers simply transitively, it is convenient to choose an arbitrary but fixed chamber Co and then define the positivity of roots as follows:
IX >O<O>IX(Co) >0 ; IJ(ELI [positivity w.r.t. Col
Then it is clear that LI splits into the disjoint union of positive roots LI + and negative roots LI-, i. e., LI = LI + U LI-, and
Definition. Let nELI+ be the subset of those positive roots whose hyperplanes have codimension one intersections with Co. Then it is clear that n is the minimal subset of LI + with Co = naelt {1);}. n is called the system of simple roots (w. r. t. Co).
Theorem (11.6). (i) Let IX, [3 E LI and p, q ~ 0 be the respective largest integers such that ([3+plX), ([3-qlX)ELI. Then ([3+jlX)ELI for -q~j~p and 2([3,IJ()/(IX,IJ() =(q-p).
(ii) Let 7rc,1+ be the system of simple roots. Then a;=I=a j E7r implies (a;,a)::::;O and 7r forms a basis of 9* such that every roots f3 E ,1 is an integral linear combination of simple roots with uniform sign, i. e.,
(iii) Let 91,92 be two simple compact Lie algebra ,11' ,12 and 7r l' 7r2 be respectively their root systems and systems of simple roots. Then 91 ~ 92=>7r 1 and 7r2 isometric, and an isometry of 7r 1,7r2 can be uniquely extended to an isometry of ,11,,12'
Proof. (i) Since ,1 is invariant under the Weyl group W, it is obvious that (f3 + pa, a)
ra(f3 + p a) = (f3 + pa) - 2 ( . a = (f3 - q a). Hence, one has 2(f3, a)/(a, a) = (q - ]i). a, a)
The fact that (f3+ja)E,1 for -q::::;j::::;p follows directly from Theorem (II.1) of ~ 1-A applies to AdG I Ga'
(ii) Observe that if a positive root a can be decomposed into the sum of two other positive roots, a=a 1 +a2 , then the condition a(h);?!O is already implied by the conditions a1(h);?!0 and az(h);?!O, and hence can be omitted from the above expression of Co as intersection of half spaces. Therefore, it is easy to see that 7r S; ,1 + are exactly those indecomposable positive roots. Let a; =1= a j E 7r be two simple roots. Then (a j - a)¢, ,1, for otherwise, either (aj - a)E ,1 + =>aj
=(aj-aj)+lXj is decomposable or (a j -a;}E,1+ and IXj=(lXj-lXj)+aj is de composable. Hence, it follows from (i) that q =0 and 2(lXj , a)!(a j , a) = (q - p) = - p::::; 0, i. e., (a j , a)::::; O. Then, it is a simple fact of linear algebra that positivity of a j and (aj,a)::::;O for all 1::::;i::::;j::::;r=>7r={a 1, ... ,ar } linearly independent. Therefore every positive (resp. negative) root f3 E ,1 + can be expressed uniquely as linear combination of a j E7r with non-negative (resp. non-positive) integral coefficients. The fact that 7r spans 9* follows easily from the fact that 9 has no center.
(iii) Since all Cartan subalgebras of a compact Lie algebra are conjugate to each other, one may modify the given isomorphism I: 91->92 by a suitable inner automorphism so that I (91) = 92' Hence it follows directly from the de finition of root system and the intrinsic inner product that I induces an isometry of ,11 onto ,12' Furthermore, since. W acts simply transitively on the set of chambers and the choice of a system of simple roots, 7r, is in 1 -1 correspondence with the choice of a chamber, it is clear that W also permutes simply transitively among different systems of simple roots. Hence, after a suitable modification by a con jugation of an element of W, we have the induced isometry maps 7r1 onto 7r2.
Finally, it is an easy consequence of (i) that ,1 is completely determined by the metric property of 7r, and hence an isometry of 7rl onto 7r2 extends uniquely to an isometry of ,11 onto ,12' 0
Dynkin diagram.
h 2(aj,a) 2(a j.a) ... . Observe t at ---) . --'- < 4 Implies that the mteger 2(a j,IX)I(a j ,a;) IS
(aj,a; (aj,a) either 0, -1, - 2 or - 3, which geometrically corresponds to the cases that the angle between a; and aj is 90", 120°, 135°, or 1500 respectively. Therefore, it is convienent to record the metric property of the system of simple roots 7r in terms of the following diagram:
Symbolically, we represent each simple root by a point and we join two points by a single, or double, or triple bond if the angle between the respective simple roots is 120°, or 135°, or 150°. Moreover, in the case of double or triple bond, i.e., 2(a i,a)/(ai,ai ) = -2, or -3 and 2(ai,a)/(aj,a) = -1, the two simple roots are not of equal length. Hence it is natural to use directed bond (~ or -) to indicate which one is longer than the other. Such a diagram is called the Dynkin diagram of the system of simple roots n (or of the Lie algebra g).
Proposition. The Dynkin diagram of a simple compact Lie algebra is connected. In general, there is a 1-1 correspondence between the connected components of its Dynkin diagram and the simple normal subalgebras of a compact Lie algebra g.
Proof. Suppose n' is a connected component of nand n = n' + nil. Then it follows from definition that n'l. nil. Hence, by the above theorem, (a',a")=O if a',a" are respectively linear combinations of n', nil, and therefore a' + a" rf,1 (for 2(a',a")/ (a',a')=(q-p)=-p=O) which in turn implies [Xa"Xa,,] =0. Then, it is easy to show that the subalgebra generated by X a" a' E < n') is a normal subalgebra and the proposition follows. 0
A connected Dynkin diagram is called geometrically feasible if there exists a set of vectors with the metric property indicated by the given diagram. Of course, for the purpose of such a purely geometric consideration, only the angles are essential and it is not difficult to see that a necessary and sufficient condition for a set of vectors {a l' ... , ar } with preassigned angles is geometrically feasible is that ILtja)2 = (2)jaj, Ltjaj)~O forany tjE1R, (and =0 only when Ltjaj=O). Hence, it is rather elementary to prove the following:
Theorem (11.7). There are only the following geometrically feasible connected Dynkin diagram:
An> n ~ 1; 0---0--0 •.. 0----0
Bn> n ~ 2; 0---0--0 ••. o~o
C n' n ~ 3; 0---0--0 .• ' 0<=0
Dn>n~4; 0----0' .'.~
E6:~;E7:~;E8:~
We refer the reader to § 5, Ch. IV of Jacobson's Lie Algebra for a standard proof of the above elementary theorem.
(C) Chevalley Basis and Classification Theorem
In the case of compact Lie algebras, the classification theorem of Cartan-Killing can be simply stated as follows:
Theorem (11.8) (Cartan-Killing). The map oj assigning a semi-simple compact Lie algebra 9 to its Dynkin diagram D(g) is a bijection between the set oj isomorphic classes oj semi-simple compact Lie algebras and the set oj all geometrically Jeasible Dynkin diagrams. (cf. Theorem (II.6).
We shall prove the above theorem as follows:
Let 9 be a semi-simple compact Lie algebra gc = 9 ® CC be its complexification. Let l) be a Cartan subalgebra and Ll be the root system of g. Then, by definition, we have the following decomposition of gc w. r. t. AdT (or adl}):
where CCa={XEgc; [H,X]=ia(H)·X for HEl}} and are one-dimensional. If one restrict the adjoint action to those subgroups, {G,,; aELl+}, next to the maximal torus T, then the above root-space decomposition are strung into invariant subspaces of AdG« (resp. adg.) as follows:
It is, then, straightforward to verify the following properties of root-spaces decomposition:
(i) For a,pELl,[CCa,CCp]=CCa+P if we set CCa+p=O for the case a+p~Ll.
Proof. [H, [Xa'Xp]] = [[H,Xa],Xp] + [Xa' [H,Xp]] =i(a+ P)(H)· [Xa'Xp] , H El}, XaECCa, XpECCp.
and thefact that a + PELl = [Xa,Xpl;60 follows from the fact that (I)= _qCCp+ jJ is irreducible w. r. 1. adg«.
(ii) For each root aELl, let H~El} be such that (H~,H)=a(H) for all HEl} and Ha=2H:J(a, a). Then each Ha is an integral linear combination of Hj =Ha; where {ai' ... , ar } =n is a system of simple roots.
Proof. Write rj for rajE W. Then·
.H~=H~ - 2(a j ,a) H'. rJ I I (a. a.) J'
J' J
and hence
since {rj} generates W, and a=w'a j for suitable WE Wand ajEn, (ii) follows.
(iii) By looking at ga® CC=l}a® CC+ga® CC=l}a® CC+({Ha} ® CC+CC.+CC_.), it is obvious that [Xa,X_a]=A.·Ha#O for Xa,X- a non-zero vectors of CCot
and CC- a respectively. Moreover, the following identity
implies that [Xa'X -a] =i(Xa'X _J·H~=!(Xa'X _J(ex,ex)iHa, (Xa,X -a)o6O. In fact, it is not difficult to show that there exist Xa'X -a such that X",=X -a and (Xa,X_a)=2/lexI2, i.e., [Xa,X_a]=iHa. [Two such pairs differ by a factor of eiO, i.e., {eiO X a , e- iO X -,J]
(iv) Let {Xa,exEL1} be so chosen that Xa=X-a and [Xa,X-a]=iHa. Define Na,p by [Xa,Xp]=N",pXa+p if (ex+P)EL1 and Na,p=O if (ex+P)rf;L1. Then one has the following properties of the structural coefficients N a, p'
(a) N -a, _p=Na,p and Na,p= -Np,a: Obvious from definition.
Na,p_ Np,y_ Ny,a (b) For a triangle of roots ex+P+y=O, one has W - lexl2 - IPI2 '
Proof. 2 ~Ig = (NaP X _y,Xy)=([Xa,Xp],Xy)
[ ] N~ =-(Xp, XaXy )=Nya(Xp,X _p)=2 IPI2 '
(c) Suppose ex,P,y,b,EL1 and ex+P+y+b=O but no two are proportional. Ibl 2
Then [Xa, [Xp,Xy]] =Npy[Xa'Xp+y] =Npy·Na,p+yX -o=N pyNoa IP +y12 . X -0'
Hence, it follows from the Jacobi identity that
2 lex+Pl 2 ( 2. (d) INapl =p(q+1) IPI2 = q+1) ,
(where ex,PEL1 and P+jexEL1, -q~j~p~1).
Proof. [X -a [Xa'Xp]] = NaP [X -aXa+p]
=Nap·N-a,a+PXp apply (b) to (-ex),(-P), (ex+P)]
IPI 2
=Nap·N -p,-a lex+PI2' Xp
_ 2 Ifil2 --INapl 'lex+PI2XP [by(a)].
On the other hand, it is a simple fact of the Ga (or rather, gJ representation on (L<Cp+ja) that [X_ a[Xa,Yp]]=-p(q+1)Yp for any YpE<Cp. Hence, one has
o -!::urthermore, let Tap~TanTp, Gap=N (TaP) and Gap = Gap/Tap, It is clear that Gap is of rank 2 and (GaP) cons~ts of all those roots of L1 w~ich are linear com bination of ex, p. Since ex + P E L1 (G aP)' the Dynkin diagram of G must be connected
and hence either 0---0, or 0=0, or 0$0. Then it is a simple matter (though a little
tedious) to check in the above three cases that p(q + 1) ICXI;I~12 = (q + 1)2.
Theorem (U.8') (Chevalley). Let 9 be a compact semi-simple Lie algebra and gc = g® <C be its complexification; gc = l) ® <C + LaELl <Ca. Then, it is possible to choose XaE<Ca such that
X a= X -a; [Xa' X -J = iHa = integral linear combination of iHaj ,
[Xa'Xp] = ±(q + l)Xa+p.
Hence the above {X a' CXE .1} together with {iH aj; cx jE n} form a basis oj' gc such that all the structural constants are integral. It is called the Chevalley basis of gcr.
Remarks (i) {Haj,CXjEn}u {(Xa+X -a)' i(Xa-X -a); cxE.1+} forms a basis of g. (ii) The above theorem clearly implies the "if part" ofthe classification theorem,
namely, .1l~.12=gl~g2 (orresp. glc~g2cl- (iii) Since the existence of Lie algebras of the classical types, i. e., An' Bn, Cn' D,P
is a well known fact, one need only to show the existence of a simple Lie algebra; for each of the five exceptional types. In view of the above explicit basis and structural constants, it is a matter of straightforward verification.
Proof of the Chevalley theorem. Since IN apl2 =(q + 1)2, one need only to show that it is possible to choose Xx so ..!hat Na,li ~re all real numbers. Note that two pairs {Xa' X -a} and {X~, X'-a} with Xa =X -a' X~ = X'-x and (Xa' X -a) =(X~, X'-a) =2/10:1 2
differ by a factor of eiO, i. e., X~ = eiO . X x and X_ a = e - iO X _ ct' It is natural to begin with an arbitrary basis {Xa'X -a; O:E.1+} with Xa=X -z and (Xa,X _a)=2/10:1 2
and then inductively adjust each pair by suitable factor of eiO to make N a. p all real. Let .1p = {O:E.1; - P < 0: < p}, pE .1+. We may assume that Na,pEIR for o:,{J,(cx + {J)E.1p and proceed to prove the induction step that N a, p E IR for 0:, {J, (0: + If) ELl p u { ± p } . If p can not be expressed as the sum of two vectors of .1 p , then we don't have to adjust {X p' X _ p}, Otherwise, let p = 0: + {J be such an expression with smallest 0:. We simply adjust {Xp,X _,,} so that N a•1i is real. (In fact, there are exact two ways by making NaP> 0 or < 0 respectively). Suppose ), + p = 0: + (J = p is another such expression. Then o:+{J+(-),)+(-p)=O and it follows from (c) of (iv) that NAIL is also real. This completes the induction step and the theorem follows by induction. D
(D) A Theorem of Weyl and the Determination of Z(G) for Simple Connected G
Theorem (H.9) (Weyl). Let G be a semi-simple compact connected Lie group. Then the simply connected (or universal) covering group G of G is also compact.
Proof Suppose the contrary. Then ker(G-->G) S Z(G) is an infinite discrete abelian subgroup of G. Hence it is easy to see that there are compact covering groups G1 ; G-->Gj-->G, with Z(G 1) of arbitrary large finite order, which clearly contradicts the fact that ord (Z (G I)) ~ the number of vertices in the Cartan polyhedron of G1 (obviously bounded). D
Finally, for the sake of reference, we list the Dynkin diagram of the Cart an polydera together with the centers of those simple, compact, simply connected Lie groups as follows:
>-"'0=>0
~ ~Z(Dn)={~:+Z2 if n even
if n odd
~~ Z(E,) Z"C(E,) ~~ Z(E,)~Z, ~ Z(Es)={id}.
Remark. (i) In the above diagram, each dot represesents a "wall" of the Cart an polyhedron, they are respectively rxiH)~O, (ljEn and f3(H)::;;' 1 where f3 is the highest root which is represented by the dark dot.
(ii) Let x be a vertex of the Cart an polyhedron and x be the opposite wall of x. Then the Dynkin diagram D(Gx ) of the centralizor of x, Gx , is exactly the one obtained by removing the dot of x from the above diagram of Cartan polyhedron. Hence, Z (G) is in 1 -1 correspondence with those vertices with C( G) - {x} = D( G).
§ 3. Classification of Irreducible Representations
(A) Classification Theorem (11.10) (Cartan-Weyl). Let G be a simply connected, semi-simple compact Lie group, 9 be its Lie algebra, l) be a Cartan subalgebra and Wbe the Weyl group of G. Also let LI be the root system and n be the system of simple roots (w. r. t. a fixed ordering). Let IjI be an irreducible complex representation of G and Q(IjI) be the weight system of 1jI. We shall denote the largest weight vector in Q(IjI) (w.r.t. the fixed ordering) by A", and call it the highest weight ofljl. Then,
(i) The multiplicity of A", is one, and any two irreducible complex representations 1jI, <p are equivalent iff their highest weights are the same, i. e., IjI ~ <p ¢> A", = A<p.
(ii) The character of IjI can be given in terms of A", by the following formula of Weyl:
I det(u) e21ti<Y(A",H)(t)
x",(t) = <Yf d t( ) 2"i<Y.l(t) , where i5=tI"EJ+ rx. <YEW e u e
(iii) A vector A E 9* can be realized as the highest weight of an irreducible complex representation iff 2(A,rx)/(rxj,rx)=qj are non-negative integers for rxjEn.
§ 3. Classification of Irreducible Representations 31
Proof Let X",(g) be the character function of Ij; and X",(t) be its restriction to the maximal torus T (with l) as its Lie algebra). Let mew) be the multiplicity of w in Q(Ij;). Then, by definition, x",(t) = Lm(w)e21tiw (l).
In view of the Weyl integration formula (cf. § 1-0), it is natural to try to deter mine the function x",(t). Q(t) =x",(tHLuewdetae21tiUdU»). Note that X",(t) is symmetric and Q(t) is anti-symmetric and hence X",(t). Q(t) is anti-symmetric (w.r.t. to W-action). If one expands an anti-symmetric function f in terms of linear combination of L2-basis of L2(T) consists of representation functions, it is easy to see that
f =" c .(" det(a) e21tiu , v(t») L.,veCo v L.,O'E W
where v runs through weight vectors in the positive Weyl chamber Co. Hence,
x",(t)· Q(t) = {m(A",)' e21tiA .,U) + ... }. {Luewdet(a) e21tiu, d(t)}
= m(A",)' Luewdet(a)· e21tiu (A.,H)(t) + possible more terms.
Now the irreducibility of Ij; (cf. Theorem (1.3'), § 1-C, Ch. I) implies that
f - 1f 2 1 2 1 = G x",(g)· x",(g)dg = - T Ix",(t)Q(t)1 dt = - IIx",(t)· Q(t)IIL2(T) w w
= ! IIm(A",)' Luewdet(a)e21tiu(A.,H)(t) + '''IIL2(T) (by Schur orthogonality)
1 f 2 } 2 =-\m(A",) ·W+"· ~m(A",) . w
Hence, one must have m(A",) = 1 and X",(t)·Q(t) = Luewdet(a)e21tiu(A.,H)(t) which is exactly the Weyl character formula. Since the character is a complete invariant and the above Weyl character formula gives an explicit expression of X",(t) in turms of the highest weight A"" it is obvious that Ij; ~ <p~ A", = A",. There fore, (i) and (ii) are completely proved; (iii) is a direct consequence of the com pleteness theorem of Peter-Weyl. 0
. n (A", +£5,0() Corollary (11.10.1). dlmlj; = ~eLl+ (£5,0() .
Proof Observe that dim Ij; = X", (id) = X",(O). However, the above formula of X",(t) reduces to a meaningless form of % if one simply substitute zero into it. Hence, we shall instead use the formula to compute
dimlj; =limt_Ox",(t).
For this purpose, it is convient to identify l)* with l) via the inner product and rewrite the Weyl formula as
L det(a)e21ti (U(A H),t) X(t) = Ldet(a)e21ti (ud,t)
Notice that
I det(a) e2rri (<1(A +<1).s") = L det(O") e2rri (,,·".s(A +<1»
= Q(s· (A + 8)) = TIaELI + 2i sin (n <1X,s(A + 8)).
(Cf. §1-D) Hence
d · ,I l' (-) l' Q(s·(A+8)) 1m.;! = Ims~oX s·(j = Ims~o Q(s.b)
-TI r sinn<lX,s(A+b)_TI <1X,A+b) 0 - aELI+ Ims~o . < ~) - aELI+ < ~) .
SIn n IX, S' u IX, u
Chapter III. An Equivariant Cohomology Theory Related to Fibre Bundle Theory
In the application of cohomology theory to the study of topological transformation groups, a natural and convenient formalism is to define an equivariant cohomology theory for the category of G-spaces which effectively reflects the cohomological behavior of both the space and the G-action. Following an idea of A. Borel [cf. B 10], we shall define the equivariant cohomology of a G-space X to be the ordinary cohomology of the total space X G of the universal bundle, X -+ X G -+ EG, with the given G-space X as its typical fibre, namely
The rationale of adopting the above equivariant cohomology theory in terms of the universal bundle construction is roughly the following:
(i) Intuitively and heuristically, the complexity of the G-action on X will be reflected in the complexity of the associated universal bundle X -+ X G -+ BG ,
e. g., the associated universal bundle is trivial if and only if the G-action on X is trivial. And the classical obstruction theory, especially the characteristic classes theory, clearly demonstrates that cohomology theory can then be used to detect the complexity of X G -+ BG , which, in turn, reflects the complexity of the G-action itself.
(ii) Technically, it is not difficult to see that such an equivariant cohomology theory not only possesses convenient formal properties but is also effectively computable.
§ 1. The Construction of H6(X) and its Formal Properties
(A) The Construction of A. Borel
Let X be a given G-space and EG -+ BG be the universal G-bundle. Then the total space X G of the associated universal bundle with X as fibre may be regarded as: the orbit space of EG x X
XG =EG xG X =(EG x X)/G
34 Chapter III. Equivariant Cohomology Theory
where the G-action is given by g·(e,x)=(eg-1,gx). Since the two projections are obviously equivariant, one has the following commutative diagram:
BG +-( _..::.",--' - XG _....::":.:..2 ---+1 X IG
Next suppose that Yis a K-space, h:G~K is a homomorphism and f:X~Y is an h-equivariant map, i.e., f(g·x)=h(g)-f(x). Then, it is

cohomology theory of topological transformation groups

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