Representation Theory of Semisimple Groups: An Overview Based on Examples (PMS-36) [With a New preface by the author ed.] 9781400883974

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Representation Theory o f Semisimple Groups

PRINCETON LANDM ARK S IN M A T H E M A T IC S A N D P H Y S I C S

Non-standard Analysis, by Abraham Robinson General Theory of Relativity, by P.A.M. Dirac Angular Momentum in Quantum Mechanics, by A. R. Edmonds Mathematical Foundations of Quantum Mechanics, by John von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar Riemannian Geometry, by Luther Pfahler Eisenhart The Classical Groups: Their Invariants and Representations, by Hermann Weyl Topology from the Differentiable Viewpoint, by John W. Milnor Algebraic Theory of Numbers, by Hermann Weyl Continuous Geometry, by John von Neumann Linear Programming and Extensions, by George B. Dantzig Operator Techniques in Atomic Spectroscopy, by Brian R. Judd The Topology of Fibre Bundles, by Norman Steenrod Mathematical Methods of Statistics, by Harald Cramer Theory of Lie Groups, by Claude Chevalley Homological Algebra, by Henri Cartan and Samuel Eilenberg PCT, Spin and Statistics, and Ali That by Raymond F. Streater and Arthur S. Wightman Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, by Jurgen Moser Representation Theory of Semisimple Groups: An Overview Based on Examples, by Anthony W. Knapp

Representation Theory o f Semisimple Groups A N O V E R V IE W B A S E D O N E X A M P L E S

With a new preface by the author

A N T H O N Y W. K N A P P

P R I N C E T O N U N I V E R S I T Y P RES S PRINCETON AN D OXFORD

Copyright © 1986 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire 0X 20 1SY All Rights Reserved Originally published as part of the Princeton Mathematical Series, No. 36 Third printing, and first Princeton Landmarks in Mathematics edition, with a new preface by the author, 2001 Library o f Congress Cataloging-in-Publication Data

Knapp, Anthony W. Representation theory of semisimple groups, an overview based on examples / with a new preface by the author Anthony W. Knapp, p. cm. — (Princeton landmarks in mathematics) Originally published: 1986. Includes bibliographical references and index. ISBN 0-691-09089-0 (pbk. : alk. paper) 1. Semisimple Lie groups. 2. Representations of groups. I. Series. QA387 .K58 2001 512'.55— dc21 2001027843 British Library Cataloging-in-Publication Data is available This book has been composed in Lasercomp Times Roman Printed on acid-free paper. www.pup.princeton.edu Printed in the United States of America P

To Susan and Sarah and William for their patience

Contents

P r eface M

to t h e

a t h e m a t ic s

P r in c e t o n L a n d m a r k s

in

E d it io n

x iii xv

P reface A

C h a p t e r I. S c o p e

§1. §2. §3. §4. §5. §6. §7.

§1. §2. §3. §4. §5. §6. §7. §8.

of the

T heory

The Classical Groups Cartan Decomposition Representations Concrete Problems in Representation Theory Abstract Theory for Compact Groups Application of the Abstract Theory to Lie Groups Problems

II. SL(2, C)

C hapter

x ix

cknow ledgm ents

R e p r e s e n t a t io n s

of

SU(2), SL(2, R),

The Unitary Trick Irreducible Finite-Dimensional Complex-Linear Representations of sl(2, C) Finite-Dimensional Representations of sl(2, C) Irreducible Unitary Representations of SL(2, C) Irreducible Unitary Representations of SL(2, R) Use of SU(1,1) Plancherel Formula Problems

3 7 10 14 14 23 24

and

28 30 31 33 35 39 41 42

C h a p t e r III. C 00 V e c t o r s a n d t h e U n i v e r s a l E n v e lo p in g A lg e b r a

§1. §2. §3. §4. §5.

Universal Enveloping Algebra Actions on Universal Enveloping Algebra C00 Vectors Garding Subspace Problems

46 50 51 55 57

C h apter

§1. §2. §3. §4. §5. §6. §7. §8. §9. §10. §11.

IV.

R e p r e s e n t a t io n s

of

§4. §5. §6. §7. §8. C

h apter

§1. §2. §3. §4. §5. §6. C

hapter

§1. §2. §3. §4. §5. §6. §7.

roups

Examples of Root Space Decompositions Roots Abstract Root Systems and Positivity Weyl Group, Algebraically Weights and Integral Forms Centalizers of Tori Theorem of the Highest Weight Verma Modules Weyl Group, Analytically Weyl Character Formula Problems

C h apter V. Structure T heory

§1. §2. §3.

C o m p a c t L ie G

fo r

N

oncom pact

Cartan Decomposition and the UnitaryTrick Iwasawa Decomposition Regular Elements, Weyl Chambers, andthe Weyl Group Other Decompositions Parabolic Subgroups Integral Formulas Borel-Weil Theorem Problems

V I.

H

o l o m o r p h ic

D

isc r e t e

60 65 72 78 81 86 89 93 100 104 109 G

roups

113 116 121 126 132 137 142 147

S e r ie s

Holomorphic Discrete Series for SU(1,1) Classical Bounded Symmetric Domains Harish-Chandra Decomposition Holomorphic Discrete Series Finiteness of an Integral Problems

150 152 153 158 161 164

V I I . I n d u c e d R e p r e s e n t a t io n s

Three Pictures Elementary Properties Bruhat Theory Formal Intertwining Operators Gindikin-Karpelevic Formula Estimates on Intertwining Operators, Part I Analytic Continuation of Intertwining Operators, Part I §8. Spherical Functions §9. Finite-Dimensional Representations andthe H function

167 169 172 174 177 181 183 185 191

§10. §11. §12. C h apter

§1. §2. §3. §4. §5. §6. §7. §8. §9. §10. §11. §12. §13. §14. §15. §16. C hapter

§1. §2. §3. §4. §5. §6. §7. §8. §9. §10. C h apter

§1. §2. §3. §4. §5.

Estimates on Intertwining Operators, Part II Tempered Representations and Langlands Quotients Problems

V III.

A d m is s ib l e R e p r e s e n t a t io n s

Motivation Admissible Representations Invariant Subspaces Framework for Studying Matrix Coefficients Harish-Chandra Homomorphism Infinitesimal Character Differential Equations Satisfied by Matrix Coefficients Asymptotic Expansions and Leading Exponents First Application: Subrepresentation Theorem Second Application: Analytic Continuation of Interwining Operators, Part II Third Application: Control of K-Finite Z(gc)-Finite Functions Asymptotic Expansions near the Walls Fourth Application: Asymptotic Size of Matrix Coefficients Fifth Application: Identification of Irreducible Tempered Representations Sixth Application: Langlands Classification of Irreducible Admissible Representations Problems

IX.

C o n s t r u c t io n

of

D

isc r e t e

G

lobal

203 205 209 215 218 223 226 234 238 239 242 247 253 258 266 276

S e r ie s

Infinitesimally Unitary Representations A Third Way of Treating Admissible Representations Equivalent Definitions of Discrete Series Motivation in General and the Construction in SU(1,1) Finite-Dimensional Spherical Representations Duality in the General Case Construction of Discrete Series Limitations on K Types Lemma on Linear Independence Problems

X.

196 198 201

281 282 284 287 300 303 309 320 328 330

C haracters

Existence Character Formulas for SL(2, IR) Induced Characters Differential Equations Analyticity on the Regular Set, Overview and Example

333 338 347 354 355

§6. §7. §8. §9. §10.

Analyticity on the Regular Set, General Case Formula on the Regular Set Behavior on the Singular Set Families of Admissible Representations Problems

C h a p t e r X I. I n t r o d u c t io n

§1. §2. §3. §4. §5. §6. §7. §8. C h apter

§1. §2. §3. §4. §5. §6. §7. §8. §9. §10. C hapter

§1. §2. §3. §4. §5. §6. §7.

to

P

lancherel

F orm ula

Constructive Proof for SU(2) Constructive Proof for SL(2, C) Constructive Proof for SL(2, R) Ingredients of Proof for General Case Scheme of Proof for General Case Properties of Ff Hirai’s Patching Conditions Problems

X II.

E x h a u s t io n

of

D

is c r e t e

P

lancherel

385 387 394 401 404 407 421 425

S e r ie s

Boundedness of Numerators of Characters Use of Patching Conditions Formula for Discrete Series Characters Schwartz Space Exhaustion of Discrete Series Tempered Distributions Limits of Discrete Series Discrete Series of M Schmid’s Identity Problems

X III.

360 368 371 374 383

426 432 436 447 452 456 460 467 473 476

F orm ula

Ideas and Ingredients Real-Rank-One Groups, Part I Real-Rank-One Groups, Part II Averaged Discrete Series Sp (2, R) General Case Problems

482 482 485 494 502 511 512

C h a p t e r X I V . Ir r e d u c ib l e T e m p e r e d R e p r e s e n t a t io n s

§1. §2. §3. §4. §5. §6. §7.

SL(2, R) from a More General Point of View Eisenstein Integrals Asymptotics of Eisenstein Integrals The rj Functions for Intertwining Operators First Irreducibility Results Normalization of Intertwining Operators and Reducibility Connection with Plancherel Formula when dim A = 1

515 520 526 535 540 543 547

§8. §9. §10. §11. §12. §13. §14. §15. §16. §17. §18. C h apter

§1. §2. §3. §4. C hapter

§1. §2. §3. §4. §5.

Harish-Chandra’s Completeness Theorem R Group Action by Weyl Group on Representations of M Multiplicity One Theorem Zuckerman Tensoring of Induced Representations Generalized Schmid Identities Inversion of Generalized Schmid Identities Complete Reduction of Induced Representations Classification Revised Langlands Classification Problems

XV.

M

in im a l

K

T ypes

Definition and Formula Inversion Problem Connection with Intertwining Operators Problems

XVI.

U

n it a r y

A

p p e n d ix

D

SL(2, U) and SL(2, C) Continuity Arguments and Complementary Series Criterion for Unitary Representations Reduction to Real Infinitesimal Character Problems

§3§4. §5. §6. §7.

of

L ie G

B: R e g u l a r S in g u l a r P o in t s

of

650 653 655 660 665

roups

Lie Algebras Structure Theory of Lie Algebras Fundamental Group and Covering Spaces Topological Groups Vector Fields and Submanifolds Lie Groups

if f e r e n t ia l

§1. §2.

626 635 641 647

R e p r e s e n t a t io n s

A p p e n d ix A: E l e m e n t a r y T h e o r y

§1. §2. §3. §4. §5. §6.

553 560 568 577 584 587 595 599 606 614 621

P

667 668 670 673 674 679 a r t ia l

E q u a t io n s

Summary of Classical One-Variable Theory Uniqueness and Analytic Continuation of Solutions in Several Variables Analog of Fundamental Matrix Regular Singularities Systems of Higher Order Leading Exponents and the Analog of the Indicial Equation Uniqueness of Representation

685 690 693 697 700 703 710

A p p e n d ix C: R o o ts G

and

R e s t r ic t e d R o o t s

for

C

l a s s ic a l

roups

§1. Complex Groups §2. Noncompact Real Groups §3. Roots vs. Restricted Roots in Noncompact Real Groups

N otes R eferences I n d e x o f N o t a t io n In d e x

713 713 715

719 747 763 767

Preface to the Princeton Landmarks in Mathematics Edition I am pleased that Princeton University Press has decided to reprint in its Landmarks in Mathematics series Representation Theory o f Semisimple Groups: An Overview Based on Examples. The original hardback edition of the book has been out of print for two years, and the book continues to be in demand. The subject matter is at least as important today as it was at the time of the book’s original publication in 1986. Two of the fields of application— automorphic forms and analysis of semisimple symmetric spaces— have un­ dergone remarkable advances, and the theory in the book has been indis­ pensable for both. Newer fields, such as Kac-Moody algebras and quantum groups, show promise of using more and more of this theory. And attempts at solving the key problem in Chapter XVI— that of finding all the irreduc­ ible unitary representations for all semisimple groups— have led to new ap­ proaches and new problems in the subjects of algebraic groups and geomet­ ric group actions. Even with all these advances, the approach taken in the hardback edition continues to be an appropriate one for learning the subject. None of the text has been changed in the Landmarks edition, and thus it remains true to this approach. A.W.K. April 2001

Preface The intention with this book is to give a survey of the representation the­ ory of semisimple Lie groups, including results and techniques, in a way that reflects the spirit of the subject, corresponds more to a person’s nat­ ural learning process, and stops at the end of a single volume. O ur approach is based on examples and has unusual ground rules. Although we insist (at least ultimately) on precisely stated theorems, we allow proofs that handle only an example. This is especially so when the example captures the idea for the general case. In fact, we prefer such a proof when the difference between the special case and the general case is merely a m atter of technique and a presentation of the technique would not contribute to the goals of the book. The reader will be confronted with a first instance of this style of proof with Proposition 1.2. In some cases later on, when the style of a proof is atypical of the subject m atter of the book, we omit the proof altogether. Another aspect of the ground rules is that we feel no compulsion to state results in maximum generality. Even when the effect is to break with tra­ dition, we are willing to define a concept narrowly. This is especially so with concepts for which one traditionally makes a wider definition and then proves as a theorem that the narrower definition gives all examples. Thus, for instance, a semisimple Lie group for us has a built-in C artan involution, whereas traditionally one proves the existence of a C artan in­ volution as a theorem; since the involution is apparent in examples, we take it as part of the definition. An essential companion to this style of writing is a careful guide to fur­ ther reading for people who are interested. The section of Notes and its accompanying References are for just this purpose— so that a reader can selectively go more deeply into an aspect of the subject at will. Twice we depart somewhat from our ground rules and proceed in a more thorough fashion. The first time is in C hapter IV with the C artanWeyl theory for compact Lie groups. The theory is applied often, and its general techniques are used frequently. The second time is in C hapter VIII and Appendix B with admissible representations. The heart of this theory consists of two brilliant papers by H arish-C handra [1960] on the role of differential equations, a fundamental contribution by Langlands

[1973] on the classification of irreducible admissible representations, and a striking application of the theory by Casselman [1975]. The original papers are unpublished manuscripts, although H arish-C handra’s have been included in his collected works and parts of all the papers have been incorporated into the books by W arner [1972b] and by Borel and Wallach [1980] and into the paper by Casselman and Milicic [1982]. Since the original papers are not otherwise widely accessible, since they have been simplified somewhat by several people, and since their content is so im portant, we have chosen to go into some detail about them. The finite-dimensional representation theory of semisimple groups is due chiefly to E. C artan and H. Weyl. The infinite-dimensional theory began with Bargmann’s treatment of SL(2, IR) in 1947 and then was dominated for many years by Harish-Chandra in the United States and by Gelfand and Naim ark in the Soviet Union. Although functional analysts such as Godement, Mackey, M autner, and Segal made early contributions to the foundations of the subject, it was Harish-Chandra, Gelfand, and N aim ark who set the tone for research by using deeper structural properties of the groups to get at explicit results in representation theory. The early work by these three leaders established the explicit determination of the Plancherel formula and the explicit description of the unitary dual as im portant initial goals. This attitude of requiring explicit results ultimately forced a more concrete approach to the subject than was possible with abstract func­ tional analysis, and the same attitude continues today. M ore recently this attitude has been refined to insist that significant results not only be ex­ plicit but also be applicable to all semisimple groups. A group-by-group analysis is rarely sufficient now: It usually does not give the required am ount of insight into the subject. To be true to the field, this book at­ tempts to communicate such attitudes and approaches, along with the results. Bruhat’s 1956 thesis was the first m ajor advance in the field by another author that was consistent with the attitudes and approaches of the three leaders. Toward 1960 other mathematicians began to make significant contributions to parts of the theory beyond the foundations, but the goals and attitudes remained. Beginning with C artan and Weyl and lasting even beyond 1960, there was a continual argument among experts about whether the subject should be approached through analysis or through algebra. Some today still take one side or the other. It is clear from history, though, that it is best to use both analysis and algebra; insight comes from each. This book reflects that philosophy. To present both viewpoints for compact groups, for example, we begin with C artan’s algebraic approach and switch abruptly to Weyl’s analytic approach in the middle. The reader will notice other instances of this philosophy in later chapters.

The author’s introduction to this subject came from a course taught by S. Helgason at M.I.T. in 1967, a seminar run with C. J. Earle, W.H.J. Fuchs, S. Halperin, O. S. Rothaus, and H.-C. W ang in 1968, a course from HarishChandra in the fall of 1968, and conversations with E. M. Stein beginning in 1968. Some of these first insights are reproduced in this book. M ore of the book comes from lectures and courses given by the author over a pe­ riod of fifteen years. There are a few new theorems and many new proofs. All of this material came together for a course at Universite Paris VII in Spring 1982, and the notes given for that course constituted a pre­ liminary edition of the present book. Prerequisites for the book are a one-semester course in Lie groups, some measure theory, some knowledge of one complex variable, and a few things about Hilbert spaces. F or the one-semester course in Lie groups, knowledge of the first four chapters of the book by Chevalley [1947] and some supplementary material on Lie algebras are appropriate; a summary of this m aterial constitutes Appendix A. In addition to these prerequisites, existence and uniqueness of H aar measure are assumed, as is the definition of a complex manifold; references are provided for this material. Other theorems are sometimes cited in the text; they are not intended as part of the prerequisites, and references are given. Beginning at a certain point in one’s mathematical career— cor­ responding roughly to the second or third year of graduate school in the United States and to the troisieme cycle in France— one rarely learns a field of mathematics by studying it from start to finish. Later courses may be given as logical progressions through a subject, but the alert instructor recognizes that the students who m aster the mathematics do not do so by mastering the logical progressions. Instead the mastery comes through studying examples, through grasping patterns, through getting a feeling for how to approach aspects of the subject, and through other intangibles. Yet our advanced mathematics books seldom reflect this reality. The subject of semisimple Lie groups is especially troublesome in this respect. It has a reputation for being both beautiful and difficult, and many mathematicians seem to want to know something about it. But it seems impossible to penetrate. A thorough logical-progression approach might require ten thousand pages. Thus the need and the opportunity are present to try a different ap­ proach. The intention is that an approach to representation theory through examples be a response to that need and opportunity. A.W.K. August 1984

Acknowledgmen ts

It is difficult to see how the writing of this book could have been finished without the help of four people who gave instruction to the author, pro­ vided missing proofs, and solved various problems of exposition. The author is truly grateful to these four— R. A. Herb, R. P. Langlands, D. A. Vogan, and N. R. Wallach— for all their help. The author appreciates also the contributions of J.-L. Clerc, K. Lai, H. Schlichtkrull, Erik Thomas, and E. van den Ban, who read extensive portions of the m anuscript and offered criticisms and corrections. O ther people who provided substantive help in large or small ways were J. Arthur, M. W. Baldoni-Silva, Y. Benoist, B. E. Blank, M. Duflo, J. P. G ourdot, J.A.C. Kolk, P. J. Sally, W. Schmid, and J. Vargas Soria. Their contributions gave continual encouragement to the author during the course of the writing. Financial support for the writing came from Universite Paris VII and the Guggenheim Foundation. Some published research of the author that is reproduced in this book was supported by the N ational Science F ounda­ tion and the Institute for Advanced Study in Princeton, New Jersey.

Representation Theory o f Semisimple Groups

CHAPTER I

Scope o f the Theory §1. The Classical Groups A linear connected reductive group is a closed connected group of real or complex matrices that is stable under conjugate transpose. A linear con­ nected semisimple group is a linear connected reductive group with finite center. To avoid having cumbersome statements of theorems, we may or may not make an exception for the trivial one-element group in these definitions. We shall denote such a group typically by G. Since G is a closed subgroup of a Lie group, it is a Lie group, by (A.107). We let g be its Lie algebra, regarded as a real Lie algebra of real or complex matrices. Inverse conju­ gate transpose, which we denote by ©, is an autom orphism of G called the C artan involution. N ote that © 2 = 1. Let K = { g e G\€>g = g}\ this is a subgroup of G that will be observed presently to be a maximal compact subgroup of G. The differential 0 of 0 at 1 is an autom orphism of g, given as negative conjugate transpose. Let f and p be the eigenspaces in g under 0 for eigen­ values + 1 and —1, respectively. Since 02 = 1, we have a Cartan decom­ position for g given by g = ! © p. Here I consists of the skew-Hermitian members of g, and p consists of the Hermitian members. Since 0 is an automorphism, we have the relations [f, f] £ f,

[I, p] £ p,

[p, p] £ f.

In particular, f is a Lie subalgebra of g. Before coming to the examples, we give two propositions about such groups and their Lie algebras, deferring the proofs to §2. The decomposi­ tion G = K exp p in the second one is called the Cartan decomposition of G. 3

Proposition 1.1. If G is linear connected semisimple, then g is semisimple. M ore generally, if G is linear connected reductive, then g = Z g ® [g, g], as a direct sum of ideals. Here Z g denotes the center of g, and the com­ m utator ideal [g, g] is semisimple. Proposition 1.2. If G is linear connected reductive, then K is compact connected and is a maximal compact subgroup of G. Its Lie algebra is !. Moreover, the m ap of K x p into G given by (fc, X ) -> k exp X is a diffeomorphism onto G. The classical groups are certain linear connected semisimple groups that are intimately connected with classical geometry. We shall list them along with a few closely related reductive groups. We divide the list into three sections— complex groups, compact groups, and real noncom pact groups. Complex groups: GL(n, C) = {nonsingular n-by-n complex matrices} SL(n, C) = { g e GL(n, C )|det g = 1} SO(n, C) = {g e SL(n, C ) |^ tr = 1} Sp(n, C) = \ g e SL(2n, C)\gtrJg = J for J = These are called the complex general linear group, complex special linear group, complex special orthogonal group, and complex symplectic group, respectively. Each of these is given by polynomial conditions imposed on entries of members of GL(n, C) and hence is a closed subgroup of GL(n, C). They are thus Lie groups. Their Lie algebras are gl(w, C) = {n-by-n complex matrices} sl(n, C) = { I e gl(n, C )|T r X = 0} so(n, C) = { X e

5

l(n, C )\X + X ir = 0}

sp(n, C ) = { I e sl(2n, C )|X trJ + J X = 0}.

Each group is stable under © .N o te that 9 is not complex linear. Connectedness requires a little argument. For SL(n, C) we can rea­ son as follows, the other groups being handled in an analogous manner. SL(n, C) acts transitively on the column vectors in C" — {0} by matrix multiplication, and the subgroup that leaves fixed the last standard basis

vector is SL(n - 1, C)

O'

C n-1

j

— SL(n — 1 , C)tx C " - 1.

By(A.108) w eobtainahom eom orphism ofSL (n, C)/(SL(n — 1, C) x € ”_1) with C" — {0}. Then we can argue inductively, using the fact (A.56.2) that H connected and G/H connected implies G connected. All the groups but GL(n, C) have finite center and thus are semisimple. For these groups the Lie algebra is actually a Lie algebra over C. This property is a reflection of the fact that the exponential m ap can be used to form charts on G that make G into a complex manifold in such a way that multiplication is holomorphic. (Cf. (A.97).) Compact groups: SO(n) = {g e SL(n, C)| gtTg = 1 and g has real entries} U(ri) = { g c G L (n , C)\gtrg = 1} SU(n) = {g

g

U(n)|det g = 1}

The first three are called the special orthogonal group, the unitary group, and the special unitary group, respectively. Each of these is bounded as a subset of C " 2 (or C 4 " 2 in the case of Sp(n)), as well as closed, and hence is compact by the Heine-Borel Theorem. Their Lie algebras are so(n) = {AT e 5 l(n, C) | X tr + X = 0 and X has real entries} u(n) = { X e gl(n, C )|X tr + X = 0} 5

u(n) = {X e u(n)|Tr X = 0}

5

p(u) = { X e u(2n)|XtrJ + J X = 0}.

Each group is fixed elementwise by 0 . In fact, U(n) is K for GL(n, C) SU(n) is K for SL(n, C) SO(n) is K for SO(n, C) Sp(n) is K for Sp(n, C). By Proposition 1.2 each of these groups is connected. Thus each of the groups is linear connected reductive. Of these, SU(n) is semisimple for n > 2, SO(n) is semisimple for n > 3, and Sp(n) is semisimple for n > 1.

The group Sp(rc) has another realization— as the n-by-n unitary group for the quaternions H. This realization will be established in the Problems at the end of the chapter. Real noncompact groups: We list G, g, K , and f for the noncomplex non­ compact classical groups. G

9

K

!

SL(n, IK) SL(n, H) SO 0 (w, n) SU(w, n) Sp(m, n) Sp(n, IR) SO*(2n)

sl(n, IR) sl(n, H) so(m, n) su(m, n) sp(m, n) sp(n, IR) so*(2n)

SO (n) Sp(n) SO(m) x SO(n) S(U(m) x U(n)) Sp(w) x Sp(n) U(n) U(n)

so(n) sp(n) so(m) © so(n) s(u(m) © u(n)) sp(m) © sp(n) u(n) u(n).

SL(n, R) and SL(n, H) refer to matrices of determ inant one with real and quaternion entries, respectively. SO 0 (m, n), SU(m, n), and Sp(m, n) are the linear isometry groups for the Herm itian form N

2

+ •••+ K

|2

- |zm+i | 2 ----------- |zm

+ n |2

defined over R, C, and H, respectively, with the subscript “0” referring to the identity component. The group K = S(U(m) x U(n)) for SU(m, n) is the subgroup of U(m) x U(n) of matrices of determ inant one. To stick strictly to our definition of “linear connected reductive group,” we ought to define SL(w, H) and Sp(m, n) as groups of complex matrices; such defini­ tions by means of complex matrices will be given in the Problems at the end of the chapter. The group Sp(n, R) is the subgroup of real matrices in Sp(n, C) and can be conjugated by a unitary matrix so as to become {g e SU(n, n)\gtTJg = J}; then SO*(2n) is the analogous group

We omit all verifications of connectedness for these groups. Direct product: The direct product G x H of two linear connected re­ ductive groups G and H is linear connected reductive when realized as G 0 . If G and H are semisimple, so is G x H. 0 H

1.2

§2. C A R T A N D E C O M P O S I T I O N

1

§2. Cartan Decomposition Our definitions are arranged so as to impose the extra structure of the Cartan involution (0 or 0) on our groups and Lie algebras. (By contrast, in the general theory one would prove the existence of 0 or 0 from some weaker definition of reductive or semisimple group.) We should check that © is determined up to isomorphism by G, and we give a result in this direction in Proposition 1.4. But first we shall introduce a useful tool, the trace form for g, and we shall use it to give quick proofs of Prop­ ositions 1.1. and 1.2. The trace form B0 for g is defined by B0(X, Y) = Tr(AT). This is a complex-valued symmetric bilinear form on g x g. With respect to B0, each ad X acts by skew transformations: fl0((ad X)Y9Z ) = - B0(Y, (ad X)Z), as we see by expanding both sides. The real part Re B0 of the trace form is a real-valued symmetric bilinear form on g x g such that each ad X acts by skew transformations. Both B0 and Re B0 are nondegenerate on g x g because B0(X, 6X) < 0 if X ^ O . (l.l) Inequality (1.1) shows that Re B0 is negative definite on ! and positive definite on p. In addition, ! and p are orthogonal with respect to R q B0. [T o see that ReTr(X7) = 0 for X skew-Hermitian and Y Hermitian, conjugate X and Y by a unitary matrix to make Y diagonal with real entries, and then compute the trace.] Applying (1.1), we can define an inner product on the real vector space 9 by < X , Y > = - R e B 0(X,OY). (1.2) From the previous paragraph, I and p are orthogonal in this inner product. This inner product allows us to define adjoints of linear transformations from g into itself, and we check that ad 9X = —(ad X)*.

(1.3)

The Killing form B for g is given in (A.20) by B(X, Y) = Tr(ad X ad Y), and (1.3) shows that - B(X, OX) = Tr((ad X)(ad X)*) > 0 unless ad X

= 0.

(1.4)

Proof o f Proposition 1.1. If G has finite center, then g has 0 center and X # 0 implies ad X ^ 0. Thus in this case, (1.4) shows that the Killing form for G is nondegenerate, and g is semisimple by C artan’s criterion for semisimplicity (A.23). F or general G we can conclude in the same way that g/Zg is semisimple. Let Z g be the “orthogonal complement” of Z g with respect to the non­ degenerate form Re B 0. Since Z g is a 0-stable ideal, so is Z g. Then the restriction of Re B 0 to Z g x Z g is nondegenerate, and it follows from (A. 19) that g = Z g © Z g. Since g/Zg is semisimple, Z g is semisimple. But then g = Z g © Z g shows Z g = [g, g]. Hence [g, g] is semisimple and g = Z g © [g, g]. Proof o f Proposition 1.2 for the classical groups. Since K = G n U(n), K is a closed subgroup of a compact group and so is compact. Since © has differential 0, K has Lie algebra f. To obtain the decomposition G = K exp p, let g be in G and write g = k exp X for the Polar Decom­ position of g as a member of GL(n, C); here k is in U (n) and exp X is positive definite Hermitian, written as the exponential of a unique Hermitian matrix X. Then @g = k exp( —X ) and hence (&g)~1g = exp 2 X is in G. If we can prove that X is in g, then exp X and k = g(exp X ) ~ 1 are in G, and the m ap (fc, X) k exp X consequently carries K x p onto G. The map is one-one by the Polar Decomposition Theorem, and it is smooth because exp and multiplication are smooth. Its inverse is built from the smooth maps g -> (0 0 )“ 1g = exp 2X, and

exp 2 X -► 2 X

(jg, X) -> g (exp X )

~ 1

X,

= k

and so is smooth. (The verification that the Jacobian determ inant of X -► exp X is nowhere 0 for X Herm itian is omitted.) Thus we want to deduce that X is in g from the fact that exp X is in G. Each classical group is the connected com ponent of the identity of the zero locus of some set of real-valued polynomials in the real and imaginary parts of the matrix entries. (For GL(n) we must first imbed the group in GL(n + 1), putting (det g) - 1 in the lower right entry.) Let us conjugate matters so that exp X is diagonal, say X = diag(a1?. . . , an) with each aj real. Since exp X and its integral powers are in G, the transformed poly­ nomials vanish at (exp X f — diag(e*ai, . . . , ekan) for every integer k. Therefore the transformed polynomials vanish at diag(etfll, . . . , etan)

(1.5)

for all real t. Since (1.5) is connected, exp t X is contained in G for all real t. Therefore X is in g by (A. 103.5). Finally we want to prove that K is maximal among compact subgroups of G. If K i is a compact subgroup properly containing K, then the de­ composition G = K exp p shows that K \ contains an element of exp p other than the identity. The eigenvalues of this element are positive and not all 1. Raising the element to powers, we see that the powers cannot all lie in a bounded set. Thus the existence of K x leads to a contradiction, and we conclude K is maximal compact. Corollary 1.3. If G is linear connected reductive, then the center Z G of G satisfies Z G = (Z G n K) exp(p n Zg). Remarks. If Z G is discrete, it follows from this result that Z G is finite. This fact explains a bit the condition “finite center” in the definition of linear connected semisimple groups. Proof. For z in Z G, write z = k exp X as in Proposition 1.2. Then (®z)_1z = exp 2 X is in Z G. By the finite-dimensional Spectral Theorem, X is a polynomial in exp 2X. Since exp 2 X commutes with G, so does X. Thus X is in Z g. Since exp X is then in Z G, k is in Z G. Proposition 1.4. If Gx and G 2 are linear connected reductive groups whose Lie algebras have C artan decompositions gi = fi © Pi and g2 = I 2 © p 2, if Gl and G 2 have compact center, and if q>:^1 -> g 2 is an isomorphism, then there is another isomorphism