Progress in Mathematics Volume 140

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

Anthony W. Knapp

Lie Groups Beyond an Introduction

Springer Science+Business Media, LLC

Anthony W. Knapp Department of Mathematics State University of New York at Stony Brook Stony Brook, NY 11794

Library of Congress Cataloging-in-Publication Data

Knapp, Anthony W., 1941-Lie groups beyond an introduction I Anthony W. Knapp.

p. em. -- (Progress in mathematics ; v. 140) Includes bibliographical references (p. - ) and index. ISBN 978-1-4757-2455-4 ISBN 978-1-4757-2453-0 (eBook) DOI 10.1007/978-1-4757-2453-0

1. Lie groups. 2. Lie algebras. 3. Representations of groups. I. Title. II. Series: Progress in mathematics (Boston, Mass.) ; vol. 140. QA387.K567 1996 512'.55--dc20

96-17002 CIP

Printed on acid-free paper m® © 1996 Anthony W. Knapp Birkhiiuser lL(J» Softcover reprint of the hardcover 1st edition 1996 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy­ing, recording, or otherwise, without prior permission of the copyright owner.

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To Susan

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Sarah and Dave and Will

CONTENTS

List of Figures X

Preface xi Prerequisites by Chapter xiv Standard Notation XV

I. LIE ALGEBRAS AND LIE GROUPS 1 1. Definitions and Examples 2 2. Ideals 7 3. Field Extensions and the Killing Form 11 4. Semidirect Products of Lie Algebras 15 5. Solvable Lie Algebras and Lie's Theorem 17 6. Nilpotent Lie Algebras and Engel's Theorem 22 7. Cartan's Criterion for Semisimplicity 24 8. Examples of Semisimple Lie Algebras 31 9. Representations of sl(2, C) 37

10. Elementary Theory of Lie Groups 43 11. Automorphisms and Derivations 55 12. Semidirect Products of Lie Groups 58 13. Nilpotent Lie Groups 62 14. Classical Semisimple Lie Groups 66 15. Problems 73

II. COMPLEX SEMISIMPLE LIE ALGEBRAS 79 1. Classical Root Space Decompositions 80 2. Existence of Cartan Subalgebras 85 3. Uniqueness of Cartan Subalgebras 92 4. Roots 94 5. Abstract Root Systems 103 6. WeylGroup 116 7. Classification of Abstract Cartan Matrices 123 8. Classification of Nonreduced Abstract Root Systems 138 9. Serre Relations 139

10. Isomorphism Theorem 149 11. Existence Theorem 152 12. Problems 156

vii

viii Contents

Ill. UNIVERSAL ENVELOPING ALGEBRA 164 1. Universal Mapping Property 164 2. Poincare-Birkho:ff-Witt Theorem 168 3. Associated Graded Algebra 172 4. Free Lie Algebras 177 5. Problems 179

IV. COMPACT LIE GROUPS 181 1. Examples of Representations 181 2. Abstract Representation Theory 186 3. Peter-Weyl Theorem 191 4. Compact Lie Algebras 196 5. Centralizers of Tori 198 6. Analytic Weyl Group 206 7. Integral Forms 210 8. Weyl's Theorem 214 9. Problems 215

v. FINITE-DIMENSIONAL REPRESENTATIONS 219 1. Weights 220 2. Theorem of the Highest Weight 225 3. Verma Modules 229 4. Complete Reducibility 236 5. Harish-Chandra Isomorphism 246 6. Weyl Character Formula 259 7. Parabolic Subalgebras 269 8. Application to Compact Lie Groups 277 9. Problems 283

VI. STRUCTURE THEORY OF SEMISIMPLE GROUPS 291 1. Existence of a Compact Real Form 292 2. Cartan Decomposition on the Lie Algebra Level 298 3. Cartan Decomposition on the Lie Group Level 304 4. lwasawa Decomposition 311 5. Uniqueness Properties of the lwasawa Decomposition 320 6. Cartan Subalgebras 326 7. Cayley Transforms 330 8. Vogan Diagrams 339 9. Complexification of a Simple Real Lie Algebra 348

10. Classification of Simple Real Lie Algebras 349 11. Restricted Roots in the Classification 362 12. Problems 367

Contents ix

VII. ADVANCED STRUCTURE THEORY 372 1. Further Properties of Compact Real Forms 373 2. Reductive Lie Groups 384 3. K AK Decomposition 396 4. Bruhat Decomposition 397 5. Structure of M 401 6. Real-Rank-One Subgroups 408 7. Parabolic Subgroups 411 8. Cartan Subgroups 424 9. Harish-Chandra Decomposition 435

10. Problems 450

VIII.INTE<iRATION 456 1. Differential Forms and Measure Zero 456 2. Haar Measure for Lie Groups 463 3. Decompositions of Haar Measure 468 4. Application to Reductive Lie Groups 472 5. Weyl Integration Formula 479 6. Problems 485

APPENDICES

A. Tensors, Filtrations, and Gradings 1. Tensor Algebra 487 2. Symmetric Algebra 492 3. Exterior Algebra 498 4. Filtrations and Gradings 501

B. Lie's Third Theorem 1. Levi Decomposition 504 2. Lie's Third Theorem 507

c. Data for Simple Lie Algebras 1. Classical Irreducible Reduced Root Systems 508 2. Exceptional Irreducible Reduced Root Systems 511 3. Classical Noncompact Simple Real Lie Algebras 518 4. Exceptional Noncompact Simple Real Lie Algebras 531

Hints for Solutions of Problems 545 Notes 565 References 585 Index of Notation 595 Index 599

LIST OF FIGURES

2.1. Root strings in .sl(3, C) and .sp(2, C) 98

2.2. Abstract root systems with V = IR2 105

2.3. Dynkin diagrams for An, Bn, Cn, Dn, Gz 114

2.4. Classification of Dynkin diagrams 136

2.5. Substitute Dynkin diagram for (BC)n 139

6.1. Association of classical matrix algebras to Vogan diagrams with the trivial automorphism 355

6.2. Noncompact noncomplex exceptional simple real Lie algebras with the trivial automorphism in the Vogan diagram 357

6.3. Noncompact noncomplex exceptional simple real Lie algebras with a nontrivial automorphism in the Vogan diagram 361

X

PREFACE

Fifty years ago Claude Chevalley revolutionized Lie theory by pub­lishing his classic Theory of Lie Groups I. Before his book Lie theory was a mixture of local and global results. As Chevalley put it, "This limitation was probably necessary as long as general topology was not yet sufficiently well elaborated to provide a solid base for a theory in the large. These days are now passed:'

Indeed, they are passed because Chevalley's book changed matters. Chevalley made global Lie groups into the primary objects of study. In his third and fourth chapters he introduced the global notion of ana­lytic subgroup, so that Lie subalgebras corresponded exactly to analytic subgroups. This correspondence is now taken as absolutely standard, and any introduction to general Lie groups has to have it at its core. Nowadays "local Lie groups" are a thing of the past; they arise only at one point in the development, and only until Chevalley's results have been stated and have eliminated the need for the local theory.

But where does the theory go from this point? Fifty years after Cheval­ley's book, there are clear topics: E. Cartan's completion ofW. Killing's work on classifying complex semisimple Lie algebras, the treatment of finite-dimensional representations of complex semisimple Lie algebras and compact Lie groups by Cartan and H. Weyl, the structure theory begun by Cartan for real semisimple Lie algebras and Lie groups, and harmonic analysis in the setting of semisimple groups as begun by Cartan and Weyl.

Since the development of these topics, an infinite-dimensional repre­sentation theory that began with the work of Weyl, von Neumann, and Wigner has grown tremendously from contributions by Gelfand, Harish­Chandra, and many other people. In addition, the theory of Lie algebras has gone in new directions, and an extensive theory of algebraic groups has developed. All of these later advances build on the structure theory, representation theory, and analysis begun by Cartan and Weyl.

With one exception all books before this one that go beyond the level of an introduction to Lie theory stick to Lie algebras, or else go in the direction of algebraic groups, or else begin beyond the fundamental "Cartan decomposition" of real semisimple Lie algebras. The one excep-

xi

xii Preface

tion is the book Helgason [ 1962], * with its later edition Helgason [ 1978]. Helgason's books follow Cartan's differential-geometry approach, developing geometry and Lie groups at the same time by geometric methods.

The present book uses Lie-theoretic methods to continue Lie theory beyond the introductory level, bridging the gap between the theory of complex semisimple Lie algebras and the theory of global real semisim­ple Lie groups and providing a solid foundation for representation theory. The goal is to understand Lie groups, and Lie algebras are regarded throughout as a tool for understanding Lie groups.

· The flavor of the book is both algebraic and analytic. As I said in a preface written in 1984, "Beginning with Cartan and Weyland 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." That statement remains true.

Examples play a key role in this subject. Experts tend to think ex­tensively in terms of examples, using them as a guide to seeing where the theory is headed and to finding theorems. Thus examples properly play a key role in this book. A feature of the presentation is that the point of view-about examples and about the theory-has to evolve as the theory develops. At the beginning one may think about a Lie group of matrices and its Lie algebra in terms of matrix entries, or in terms of conditions on matrices. But soon it should no longer be necessary to work with the actual matrices. By the time one gets to the last two chapters, the point of view is completely different. One has a large stock of examples, but particular features of them are what stand out. These features may be properties of an underlying root system, or relationships among subgroups, or patterns among different groups, but they are far from properties of concrete matrices.

A reader who wants only a limited understanding of the examples and the evolving point of view can just read the text. But a better under­standing comes from doing problems, and each chapter contains some in its last section. Some of these are really theorems, some are examples

·that show the degree to which hypotheses can be stretched, and some are exercises. Hints for solutions, and in many cases complete solutions, appear in a section near the end of the book. The theory in the text never relies on a problem from an earlier chapter, and proofs of theorems in the text are never left as problems at the end of the current chapter.

*A name followed by a bracketed year points to the list of References at the end of the book.

Preface xiii

A section called Notes near the end of the book provides histori­cal commentary, gives bibliographical citations, tells about additional results, and serves as a guide to further reading.

The main prerequisite for reading this book is a familiarity with elementary Lie theory, as in Chapter N of Chevalley [1946] or other sources listed at the end of the Notes for Chapter I. This theory itself requires a modest amount of linear algebra and group theory, some point-set topology, the theory of covering spaces, the theory of smooth manifolds, and some easy facts about topological groups. Except in the case of the theory of involutive distributions, the treatments of this other material in many recent books are more consistent with the present book than is Chevalley's treatment. A little Lebesgue integration plays a role in Chapter N. In addition, existence and uniqueness of Haar measure on compact Lie groups are needed for Chapter N; one can take these results on faith or one can know them from differential geometry or from integration theory. Differential forms and more extensive integration theory are used in Chapter Vlll. Occasionally some other isolated result from algebra or analysis is needed; references are given in such cases.

Individual chapters in the book usually depend on only some of the earlier chapters. Details of this dependence are given on page xiv.

My own introduction to this subject came from courses by B. Kostant and S. Helgason at MJ.T. in 1965-67, and parts of those courses have heavily influenced parts of the book. Most of the book is based on various courses I taught at Cornell University or SUNY Stony Brook between 1971 and 1995. IamindebtedtoR.Donley,J.J.Duistermaat,S. Greenleaf, S. Helgason, D. Vogan, and A. Weinstein for help with various aspects of the book and to the Institut Mittag-Leffler for its hospitality during the last period in which the book was written. The typesetting was by A_MS-TJ3)C, and the figures were drawn with Mathematic~.

May 1996

PREREQillSITES BY CHAPTER

This book assumes knowledge of a modest amount of linear algebra and group theory, some point-set topology, the theory of covering spaces, the theory of smooth manifolds, and some easy facts about topological groups. The main prerequisite is a familiarity with elementary Lie the­ory, as in Chapter IV of Chevalley [1946]. The dependence of chapters on earlier chapters, as well as additional prerequisites for particular chapters, are listed here.

CHAPTER I. Tensor products of vector spaces ( cf. § 1 of Appendix A). CHAPTER ll. Chapter I. Starting in §9: The proof of Proposition 2.96

is deferred to Chapter ill, where the result is restated and proved as Proposition 3 29. Starting in § 11: Tensor algebra as in § 1 of Appendix A.

CHAPTER ill. Chapter I, all of Appendix A. CHAPTER IV. Chapter I, tensor and exterior algebras as in §§1-3

of Appendix A, a small amount of Lebesgue integration, existence of Haar measure for compact groups. The proof of Theorem 420 uses the Hilbert-Schmidt Theorem from functional analysis. Starting in §5: Chapter fl.

CHAPTER V. Chapters ll, ill, and IV. The proof of Theoreem 5.62 uses the Hilbert Nullstellensatz.

CHAPTER VI. Chapters ll and IV. CHAPTER Vll. Chapter VI. Starting in §5: Chapter V. Starting in §8:

complex manifolds (apart from complex Lie groups). CHAPTER vm. Chapter vn, differential forms, more Lebesgue

integration. APPENDIX B. Chapters I and V.

xiv

Item

#S or lSI 0 Ec

8ij n positive Z,Q>,R,C Re z,lm z z 1 1 or I dimV V* R_n,cn

TrA detA A' A* A diagonable

diag(a!' ... 'an) EndV GL(V) [A: B] E9V; span(S) Go ZA(B) NA(B) coo

STANDARD NOTATION

Meaning

number of elements in S empty set complement of set, contragredient module lifi=j,Oifi=tfj n>O integers, rationals, reals, complex numbers real and imaginary parts of z complex conjugate of z multiplicative identity identity matrix dimension of vector space dual of vector space spaces of column vectors trace of A determinant of A transpose of A conjugate transpose of A A has a basis of eigenvectors with eigenvalues in the given field diagonal matrix linear maps of V into itself invertible linear maps of V into itself index or multiplicity of B in A direct sum of the V; linear span of S identity component of group G centralizer of B in A normalizer of B in A infinitely differentiable

Notation introduced in Appendix A and used throughout the book is generally defined at its first occurrence and appears in the Index of Notation at the end of the book.

XV