Editorial Board S. Axler K.A. Ribet - School of Mathematicsv1ranick/papers/goodwallx.pdf · Based...

Graduate Texts in Mathematics 255

Editorial Board S. Axler

K.A. Ribet

For other titles published in this series, go to http://www.springer.com/series/136

Roe Goodman Nolan R. Wallach

Symmetry, Representations, and Invariants

123

Roe Goodman Nolan R. Wallach Department of Mathematics Department of Mathematics Rutgers University University of California, San Diego Piscataway, NJ 08854-8019 La Jolla, CA 92093 USA USA [email protected] [email protected] Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California at Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA [email protected] [email protected]

ISBN 978-0-387-79851-6 e-ISBN 978-0-387-79852-3 DOI 10.1007/978-0-387-79852-3 Springer Dordrecht Heidelberg London New York

Mathematics Subject Classification (2000): 20G05, 14L35, 14M17, 17B10, 20C30, 20G20, 22E10, 22E46, 53B20, 53C35, 57M27 © Roe Goodman and Nolan R. Wallach 2009

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Based on Representations and Invariants of the Classical Groups, Roe Goodman and Nolan R. Wallach, Cambridge University Press, 1998, third corrected printing 2003.

Library of Congress Control Number: 2009927015

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Organization and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Lie Groups and Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 General and Special Linear Groups . . . . . . . . . . . . . . . . . . . . . 11.1.2 Isometry Groups of Bilinear Forms . . . . . . . . . . . . . . . . . . . . . 31.1.3 Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Quaternionic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 The Classical Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.1 General and Special Linear Lie Algebras . . . . . . . . . . . . . . . . 131.2.2 Lie Algebras Associated with Bilinear Forms . . . . . . . . . . . . . 141.2.3 Unitary Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.4 Quaternionic Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.5 Lie Algebras Associated with Classical Groups . . . . . . . . . . . 171.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Closed Subgroups of GL(n,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.3 Lie Algebra of a Closed Subgroup of GL(n,R) . . . . . . . . . . . 231.3.4 Lie Algebras of the Classical Groups . . . . . . . . . . . . . . . . . . . . 251.3.5 Exponential Coordinates on Closed Subgroups . . . . . . . . . . . 281.3.6 Differentials of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 301.3.7 Lie Algebras and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 311.3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.4.2 Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.4.3 Lie Algebra of an Algebraic Group . . . . . . . . . . . . . . . . . . . . . 39

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1.4.4 Algebraic Groups as Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 431.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.5 Rational Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.5.2 Differential of a Rational Representation . . . . . . . . . . . . . . . . . 491.5.3 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.6 Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.6.1 Rational Representations of C . . . . . . . . . . . . . . . . . . . . . . . . . 551.6.2 Rational Representations of C× . . . . . . . . . . . . . . . . . . . . . . . . 571.6.3 Jordan–Chevalley Decomposition . . . . . . . . . . . . . . . . . . . . . . 581.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1.7 Real Forms of Complex Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . 621.7.1 Real Forms and Complex Conjugations . . . . . . . . . . . . . . . . . . 621.7.2 Real Forms of the Classical Groups . . . . . . . . . . . . . . . . . . . . . 651.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2 Structure of Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.1 Semisimple Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.1.1 Toral Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.1.2 Maximal Torus in a Classical Group . . . . . . . . . . . . . . . . . . . . 722.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.2 Unipotent Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.2.1 Low-Rank Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.2.2 Unipotent Generation of Classical Groups . . . . . . . . . . . . . . . 792.2.3 Connected Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.3 Regular Representations of SL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . 832.3.1 Irreducible Representations of sl(2,C) . . . . . . . . . . . . . . . . . . 842.3.2 Irreducible Regular Representations of SL(2,C) . . . . . . . . . . 862.3.3 Complete Reducibility of SL(2,C) . . . . . . . . . . . . . . . . . . . . . 882.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.4 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.4.1 Roots with Respect to a Maximal Torus . . . . . . . . . . . . . . . . . 912.4.2 Commutation Relations of Root Spaces . . . . . . . . . . . . . . . . . . 952.4.3 Structure of Classical Root Systems . . . . . . . . . . . . . . . . . . . . . 992.4.4 Irreducibility of the Adjoint Representation . . . . . . . . . . . . . . 1042.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.5 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.5.1 Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.5.2 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.5.3 Geometry of Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.5.4 Conjugacy of Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . 1222.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Contents vii

2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3 Highest-Weight Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273.1 Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.1.1 Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.1.2 Root Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.1.3 Weight Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.1.4 Dominant Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.2 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.2.1 Theorem of the Highest Weight . . . . . . . . . . . . . . . . . . . . . . . . 1483.2.2 Weights of Irreducible Representations . . . . . . . . . . . . . . . . . . 1533.2.3 Lowest Weights and Dual Representations . . . . . . . . . . . . . . . 1573.2.4 Symplectic and Orthogonal Representations . . . . . . . . . . . . . . 1583.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.3 Reductivity of Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.3.1 Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.3.2 Casimir Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.3.3 Algebraic Proof of Complete Reducibility . . . . . . . . . . . . . . . 1693.3.4 The Unitarian Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4 Algebras and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.1 Representations of Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . 175

4.1.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.1.2 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.1.3 Jacobson Density Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.1.4 Complete Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.1.5 Double Commutant Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.1.6 Isotypic Decomposition and Multiplicities . . . . . . . . . . . . . . . 1844.1.7 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4.2 Duality for Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.2.1 General Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.2.2 Products of Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.2.3 Isotypic Decomposition of O[G] . . . . . . . . . . . . . . . . . . . . . . . . 1994.2.4 Schur–Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.2.5 Commuting Algebra and Highest-Weight Vectors . . . . . . . . . 2034.2.6 Abstract Capelli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

4.3 Group Algebras of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.3.1 Structure of Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.3.2 Schur Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . 2084.3.3 Fourier Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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4.3.4 The Algebra of Central Functions . . . . . . . . . . . . . . . . . . . . . . 2114.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

4.4 Representations of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.4.1 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.4.2 Characters of Induced Representations . . . . . . . . . . . . . . . . . . 2174.4.3 Standard Representation of Sn . . . . . . . . . . . . . . . . . . . . . . . . . 2184.4.4 Representations of Sk on Tensors . . . . . . . . . . . . . . . . . . . . . . 2214.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5 Classical Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.1 Polynomial Invariants for Reductive Groups . . . . . . . . . . . . . . . . . . . . 226

5.1.1 The Ring of Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.1.2 Invariant Polynomials for Sn . . . . . . . . . . . . . . . . . . . . . . . . . . 2285.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.2 Polynomial Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2375.2.1 First Fundamental Theorems for Classical Groups . . . . . . . . . 2385.2.2 Proof of a Basic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2425.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

5.3 Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.3.1 Tensor Invariants for GL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475.3.2 Tensor Invariants for O(V ) and Sp(V ) . . . . . . . . . . . . . . . . . . . 2485.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

5.4 Polynomial FFT for Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 2565.4.1 Invariant Polynomials as Tensors . . . . . . . . . . . . . . . . . . . . . . . 2565.4.2 Proof of Polynomial FFT for GL(V ) . . . . . . . . . . . . . . . . . . . . 2585.4.3 Proof of Polynomial FFT for O(V ) and Sp(V ) . . . . . . . . . . . . 258

5.5 Irreducible Representations of Classical Groups . . . . . . . . . . . . . . . . . 2595.5.1 Skew Duality for Classical Groups . . . . . . . . . . . . . . . . . . . . . . 2595.5.2 Fundamental Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 2685.5.3 Cartan Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2725.5.4 Irreducible Representations of GL(V ) . . . . . . . . . . . . . . . . . . . 2735.5.5 Irreducible Representations of O(V ) . . . . . . . . . . . . . . . . . . . . 2755.5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

5.6 Invariant Theory and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785.6.1 Duality and the Weyl Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 2785.6.2 GL(n)–GL(k) Schur–Weyl Duality . . . . . . . . . . . . . . . . . . . . . 2825.6.3 O(n)–sp(k) Howe Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2845.6.4 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2875.6.5 Sp(n)–so(2k) Howe Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 2905.6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

5.7 Further Applications of Invariant Theory . . . . . . . . . . . . . . . . . . . . . . 2935.7.1 Capelli Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.7.2 Decomposition of S(S2(V )) under GL(V ) . . . . . . . . . . . . . . . 2955.7.3 Decomposition of S(

∧2(V )) under GL(V ) . . . . . . . . . . . . . . . 296

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5.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2985.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

6 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

6.1.1 Construction of Cliff(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.1.2 Spaces of Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3036.1.3 Structure of Cliff(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3076.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

6.2 Spin Representations of Orthogonal Lie Algebras . . . . . . . . . . . . . . . . 3116.2.1 Embedding so(V ) in Cliff(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . 3126.2.2 Spin Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3146.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

6.3 Spin Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3166.3.1 Action of O(V ) on Cliff(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3166.3.2 Algebraically Simply Connected Groups . . . . . . . . . . . . . . . . . 3216.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

6.4 Real Forms of Spin(n,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3236.4.1 Real Forms of Vector Spaces and Algebras . . . . . . . . . . . . . . . 3236.4.2 Real Forms of Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 3256.4.3 Real Forms of Pin(n) and Spin(n) . . . . . . . . . . . . . . . . . . . . . . 3256.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

7 Character Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3297.1 Character and Dimension Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

7.1.1 Weyl Character Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3297.1.2 Weyl Dimension Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3347.1.3 Commutant Character Formulas . . . . . . . . . . . . . . . . . . . . . . . . 3377.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

7.2 Algebraic Group Approach to the Character Formula . . . . . . . . . . . . . 3427.2.1 Symmetric and Skew-Symmetric Functions . . . . . . . . . . . . . . 3427.2.2 Characters and Skew-Symmetric Functions . . . . . . . . . . . . . . 3447.2.3 Characters and Invariant Functions . . . . . . . . . . . . . . . . . . . . . 3467.2.4 Casimir Operator and Invariant Functions . . . . . . . . . . . . . . . . 3477.2.5 Algebraic Proof of the Weyl Character Formula . . . . . . . . . . . 3527.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7.3 Compact Group Approach to the Character Formula . . . . . . . . . . . . . 3547.3.1 Compact Form and Maximal Compact Torus . . . . . . . . . . . . . 3547.3.2 Weyl Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3567.3.3 Fourier Expansions of Skew Functions . . . . . . . . . . . . . . . . . . 3587.3.4 Analytic Proof of the Weyl Character Formula . . . . . . . . . . . . 3607.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

7.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

x Contents

8 Branching Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3638.1 Branching for Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

8.1.1 Statement of Branching Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 3648.1.2 Branching Patterns and Weight Multiplicities . . . . . . . . . . . . . 3668.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

8.2 Branching Laws from Weyl Character Formula . . . . . . . . . . . . . . . . . . 3708.2.1 Partition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3708.2.2 Kostant Multiplicity Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 3718.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

8.3 Proofs of Classical Branching Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 3738.3.1 Restriction from GL(n) to GL(n−1) . . . . . . . . . . . . . . . . . . . 3738.3.2 Restriction from Spin(2n+1) to Spin(2n) . . . . . . . . . . . . . . . 3758.3.3 Restriction from Spin(2n) to Spin(2n−1) . . . . . . . . . . . . . . . 3788.3.4 Restriction from Sp(n) to Sp(n−1) . . . . . . . . . . . . . . . . . . . . 379

8.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9 Tensor Representations of GL(V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3879.1 Schur–Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

9.1.1 Duality between GL(n) and Sk . . . . . . . . . . . . . . . . . . . . . . . . 3889.1.2 Characters of Sk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.1.3 Frobenius Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3949.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

9.2 Dual Reductive Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3999.2.1 Seesaw Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3999.2.2 Reciprocity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4019.2.3 Schur–Weyl Duality and GL(k)–GL(n) Duality . . . . . . . . . . 4059.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

9.3 Young Symmetrizers and Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . 4079.3.1 Tableaux and Symmetrizers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4079.3.2 Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4129.3.3 Standard Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4149.3.4 Projections onto Isotypic Components . . . . . . . . . . . . . . . . . . . 4169.3.5 Littlewood–Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 4189.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

9.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

10 Tensor Representations of O(V) and Sp(V) . . . . . . . . . . . . . . . . . . . . . . . 42510.1 Commuting Algebras on Tensor Spaces . . . . . . . . . . . . . . . . . . . . . . . . 425

10.1.1 Centralizer Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42610.1.2 Generators and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43210.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

10.2 Decomposition of Harmonic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 43510.2.1 Harmonic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43510.2.2 Harmonic Extreme Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43610.2.3 Decomposition of Harmonics for Sp(V ) . . . . . . . . . . . . . . . . . 440

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10.2.4 Decomposition of Harmonics for O(2l +1) . . . . . . . . . . . . . . 44210.2.5 Decomposition of Harmonics for O(2l) . . . . . . . . . . . . . . . . . 44610.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

10.3 Riemannian Curvature Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45110.3.1 The Space of Curvature Tensors . . . . . . . . . . . . . . . . . . . . . . . . 45310.3.2 Orthogonal Decomposition of Curvature Tensors . . . . . . . . . . 45510.3.3 The Space of Weyl Curvature Tensors . . . . . . . . . . . . . . . . . . . 45810.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

10.4 Invariant Theory and Knot Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 46110.4.1 The Braid Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46110.4.2 Orthogonal Invariants and the Yang–Baxter Equation . . . . . . 46310.4.3 The Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46410.4.4 The Jones Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46910.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

10.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

11 Algebraic Groups and Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . 47911.1 General Properties of Linear Algebraic Groups . . . . . . . . . . . . . . . . . . 479

11.1.1 Algebraic Groups as Affine Varieties . . . . . . . . . . . . . . . . . . . . 47911.1.2 Subgroups and Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 48111.1.3 Group Structures on Affine Varieties . . . . . . . . . . . . . . . . . . . . 48411.1.4 Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48511.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

11.2 Structure of Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49111.2.1 Commutative Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . 49111.2.2 Unipotent Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49311.2.3 Connected Algebraic Groups and Lie Groups . . . . . . . . . . . . . 49611.2.4 Simply Connected Semisimple Groups . . . . . . . . . . . . . . . . . . 49711.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

11.3 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50011.3.1 G-Spaces and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50011.3.2 Flag Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50111.3.3 Involutions and Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . 50611.3.4 Involutions of Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . 50711.3.5 Classical Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 51011.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

11.4 Borel Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51911.4.1 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51911.4.2 Lie–Kolchin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52011.4.3 Structure of Connected Solvable Groups . . . . . . . . . . . . . . . . . 52211.4.4 Conjugacy of Borel Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 52411.4.5 Centralizer of a Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52511.4.6 Weyl Group and Regular Semisimple Conjugacy Classes . . . 52611.4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

11.5 Further Properties of Real Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

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11.5.1 Groups with a Compact Real Form . . . . . . . . . . . . . . . . . . . . . 53111.5.2 Polar Decomposition by a Compact Form . . . . . . . . . . . . . . . . 536

11.6 Gauss Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53811.6.1 Gauss Decomposition of GL(n,C) . . . . . . . . . . . . . . . . . . . . . 53811.6.2 Gauss Decomposition of an Algebraic Group . . . . . . . . . . . . . 54011.6.3 Gauss Decomposition for Real Forms . . . . . . . . . . . . . . . . . . . 54111.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

12.1 Some General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54512.1.1 Isotypic Decomposition of O[X ] . . . . . . . . . . . . . . . . . . . . . . . . 54612.1.2 Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54812.1.3 Function Models for Irreducible Representations . . . . . . . . . . 54912.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

12.2 Multiplicity-Free Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55112.2.1 Multiplicities and B-Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55212.2.2 B-Eigenfunctions for Linear Actions . . . . . . . . . . . . . . . . . . . . 55312.2.3 Branching from GL(n) to GL(n−1) . . . . . . . . . . . . . . . . . . . . 55412.2.4 Second Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . 55712.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

12.3 Regular Functions on Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . 56612.3.1 Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56612.3.2 Examples of Iwasawa Decompositions . . . . . . . . . . . . . . . . . . 57512.3.3 Spherical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58512.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

12.4 Isotropy Representations of Symmetric Spaces . . . . . . . . . . . . . . . . . . 58812.4.1 A Theorem of Kostant and Rallis . . . . . . . . . . . . . . . . . . . . . . . 58912.4.2 Invariant Theory of Reflection Groups . . . . . . . . . . . . . . . . . . . 59012.4.3 Classical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59312.4.4 Some Results from Algebraic Geometry . . . . . . . . . . . . . . . . . 59712.4.5 Proof of the Kostant–Rallis Theorem . . . . . . . . . . . . . . . . . . . . 60112.4.6 Some Remarks on the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 60512.4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

A Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611A.1 Affine Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

A.1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611A.1.2 Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615A.1.3 Products of Affine Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616A.1.4 Principal Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617A.1.5 Irreducible Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617A.1.6 Transcendence Degree and Dimension . . . . . . . . . . . . . . . . . . 619A.1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

12 Representations on Spaces of Regular Functions . . . . . . . . . . . . . . . . . . 545

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A.2 Maps of Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622A.2.1 Rational Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622A.2.2 Extensions of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 623A.2.3 Image of a Dominant Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626A.2.4 Factorization of a Regular Map . . . . . . . . . . . . . . . . . . . . . . . . . 626A.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

A.3 Tangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628A.3.1 Tangent Space and Differentials of Maps . . . . . . . . . . . . . . . . 628A.3.2 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630A.3.3 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630A.3.4 Differential Criterion for Dominance . . . . . . . . . . . . . . . . . . . . 632

A.4 Projective and Quasiprojective Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 635A.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635A.4.2 Products of Projective Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637A.4.3 Regular Functions and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

B Linear and Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643B.1 Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

B.1.1 Primary Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643B.1.2 Additive Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 644B.1.3 Multiplicative Jordan Decomposition . . . . . . . . . . . . . . . . . . . 645

B.2 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645B.2.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646B.2.2 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647B.2.3 Symmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650B.2.4 Alternating Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653B.2.5 Determinants and Gauss Decomposition . . . . . . . . . . . . . . . . . 654B.2.6 Pfaffians and Skew-Symmetric Matrices . . . . . . . . . . . . . . . . . 656B.2.7 Irreducibility of Determinants and Pfaffians . . . . . . . . . . . . . . 659

C Associative Algebras and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 661C.1 Some Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

C.1.1 Filtered and Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 661C.1.2 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663C.1.3 Symmetric Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663C.1.4 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666C.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

C.2 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668C.2.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668C.2.2 Poincaré–Birkhoff–Witt Theorem . . . . . . . . . . . . . . . . . . . . . . 670C.2.3 Adjoint Representation of Enveloping Algebra . . . . . . . . . . . . 672

xiv Contents

D Manifolds and Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675D.1 C∞ Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

D.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675D.1.2 Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680D.1.3 Differential Forms and Integration . . . . . . . . . . . . . . . . . . . . . . 683D.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

D.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687D.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687D.2.2 Lie Algebra of a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688D.2.3 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691D.2.4 Integration on Lie Groups and Homogeneous Spaces . . . . . . 692D.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

Preface

Symmetry, in the title of this book, should be understood as the geometry of Lie(and algebraic) group actions. The basic algebraic and analytic tools in the studyof symmetry are representation and invariant theory. These three threads are pre-cisely the topics of this book. The earlier chapters can be studied at several lev-els. An advanced undergraduate or beginning graduate student can learn the theoryfor the classical groups using only linear algebra, elementary abstract algebra, andadvanced calculus, with further exploration of the key examples and concepts inthe numerous exercises following each section. The more sophisticated reader canprogress through the first ten chapters with occasional forward references to Chap-ter 11 for general results about algebraic groups. This allows great flexibility in theuse of this book as a course text. The authors have used various chapters in a varietyof courses; we suggest ways in which courses can be based on the book later in thispreface. Finally, we have taken care to make the main theorems and applicationsmeaningful for the reader who wishes to use the book as a reference to this vastsubject.

The authors are gratified that their earlier text, Representations and Invariants ofthe Classical Groups [56], was well received. The present book has the same aim: anentry into the powerful techniques of Lie and algebraic group theory. The parts of theprevious book that have withstood the authors’ many revisions as they lectured fromits material have been retained; these parts appear here after substantial rewritingand reorganization. The first four chapters are, in large part, newly written and offera more direct and elementary approach to the subject. Several of the later parts ofthe book are also new. While we continue to look upon the classical groups as bothfundamental in their own right and as important examples for the general theory, theresults are now stated and proved in their natural generality. These changes justifythe more accurate new title for the present book.

We have taken special care to make the book readable at many levels of detail.A reader desiring only the statement of a pertinent result can find it through thetable of contents and index, and then read and study it through the examples of itsuse that are generally given. A more serious reader wishing to delve into a proof ofthe result can read in detail a more computational proof that uses special properties

xv

xvi Preface

of the classical groups, or, perhaps in a second reading, the proof in the generalcase (with occasional forward references to results from later chapters). Usually,there is a third possibility of a proof using analytic methods. Some material in theearlier book, although important in its own right, has been eliminated or replaced.There are new proofs of some of the key results of the theory such as the theoremof the highest weight, the theorem on complete reducibility, the duality theorem,and the Weyl character formula. We hope that our new presentation will make thesefundamental tools more accessible.

The last two chapters of the book develop, via a basic introduction to complexalgebraic groups, what has come to be called geometric invariant theory. This in-cludes the notion of quotient space and the representation-theoretic analysis of theregular functions on a space with an algebraic group action. A full description of thematerial covered in the book is given later in the preface.

When our earlier text appeared there were few other introductions to the area.The most prominent included the fundamental text of Hermann Weyl, The ClassicalGroups: Their Invariants and Representations [164] and Chevalley’s The Theory ofLie groups I [33], together with the more recent text Lie Algebras by Humphreys[76]. These remarkable volumes should be on the bookshelf of any serious student ofthe subject. In the interim, several other texts have appeared that cover, for the mostpart, the material in Chevalley’s classic with extensions of his analytic group theoryto Lie group theory and that also incorporate much of the material in Humphrey’stext. Two books with a more substantial overlap but philosophically very differentfrom ours are those by Knapp [86] and Procesi [123]. There is much for a studentto learn from both of these books, which give an exposition of Weyl’s methods ininvariant theory that is different in emphasis from our book. We have developedthe combinatorial aspects of the subject as consequences of the representations andinvariants of the classical groups. In Hermann Weyl (and the book of Procesi) theopposite route is followed: the representations and invariants of the classical groupsrest on a combinatorial determination of the representations of the symmetric group.Knapp’s book is more oriented toward Lie group theory.

OrganizationThe logical organization of the book is illustrated in the chapter and section depen-dency chart at the end of the preface. A chapter or section listed in the chart dependson the chapters to which it is connected by a horizontal or rising line. This chart hasa central spine; to the right are the more geometric aspects of the subject and on theleft the more algebraic aspects. There are several intermediate terminal nodes in thischart (such as Sections 5.6 and 5.7, Chapter 6, and Chapters 9–10) that can serve asgoals for courses or self study.

Chapter 1 gives an elementary approach to the classical groups, viewed either asLie groups or algebraic groups, without using any deep results from differentiablemanifold theory or algebraic geometry. Chapter 2 develops the basic structure ofthe classical groups and their Lie algebras, taking advantage of the defining repre-sentations. The complete reducibility of representations of sl(2,C) is established bya variant of Cartan’s original proof. The key Lie algebra results (Cartan subalge-

Preface xvii

bras and root space decomposition) are then extended to arbitrary semisimple Liealgebras.

Chapter 3 is devoted to Cartan’s highest-weight theory and the Weyl group. Wegive a new algebraic proof of complete reducibility for semisimple Lie algebrasfollowing an argument of V. Kac; the only tools needed are the complete reducibilityfor sl(2,C) and the Casimir operator. The general treatment of associative algebrasand their representations occurs in Chapter 4, where the key result is the generalduality theorem for locally regular representations of a reductive algebraic group.The unifying role of the duality theorem is even more prominent throughout thebook than it was in our previous book.

The machinery of Chapters 1–4 is then applied in Chapter 5 to obtain the prin-cipal results in classical representations and invariant theory: the first fundamentaltheorems for the classical groups and the application of invariant theory to represen-tation theory via the duality theorem.

Chapters 6, on spinors, follows the corresponding chapter from our previousbook, with some corrections and additional exercises. For the main result in Chap-ter 7—the Weyl character formula—we give a new algebraic group proof using theradial component of the Casimir operator (replacing the proof via Lie algebra co-homology in the previous book). This proof is a differential operator analogue ofWeyl’s original proof using compact real forms and the integration formula, whichwe also present in detail. The treatment of branching laws in Chapter 8 follows thesame approach (due to Kostant) as in the previous book.

Chapters 9–10 apply all the machinery developed in previous chapters to analyzethe tensor representations of the classical groups. In Chapter 9 we have added adiscussion of the Littlewood–Richardson rule (including the role of the GL(n,C)branching law to reduce the proof to a well-known combinatorial construction). Wehave removed the partial harmonic decomposition of tensor space under orthogonaland symplectic groups that was treated in Chapter 10 of the previous book, andreplaced it with a representation-theoretic treatment of the symmetry properties ofcurvature tensors for pseudo-Riemannian manifolds.

The general study of algebraic groups over C and homogeneous spaces beginsin Chapter 11 (with the necessary background material from algebraic geometry inAppendix A). In Lie theory the examples are, in many cases, more difficult than thegeneral theorems. As in our previous book, every new concept is detailed with itsmeaning for each of the classical groups. For example, in Chapter 11 every classi-cal symmetric pair is described and a model is given for the corresponding affinevariety, and in Chapter 12 the (complexified) Iwasawa decomposition is worked outexplicitly. Also in Chapter 12 a proof of the celebrated Kostant–Rallis theorem forsymmetric spaces is given and every implication for the invariant theory of classicalgroups is explained.

This book can serve for several different courses. An introductory one-termcourse in Lie groups, algebraic groups, and representation theory with emphasison the classical groups can be based on Chapters 1–3 (with reference to AppendixD as needed). Chapters 1–3 and 11 (with reference to Appendix A as needed) canbe the core of a one-term introductory course on algebraic groups in characteris-

xviii Preface

tic zero. For students who have already had an introductory course in Lie algebrasand Lie groups, Chapters 3 and 4 together with Chapters 6–10 contain ample mate-rial for a second course emphasizing representations, character formulas, and theirapplications. An alternative (more advanced) second-term course emphasizing thegeometric side of the subject can be based on topics from Chapters 3, 4, 11, and 12.A year-long course on representations and classical invariant theory along the linesof Weyl’s book would follow Chapters 1–5, 7, 9, and 10. The exercises have beenrevised and many new ones added (there are now more than 350, most with severalparts and detailed hints for solution). Although none of the exercises are used inthe proofs of the results in the book, we consider them an essential part of coursesbased on this book. Working through a significant number of the exercises helps astudent learn the general concepts, fine structure, and applications of representationand invariant theory.

AcknowledgmentsIn the end-of-chapter notes we have attempted to give credits for the results in thebook and some idea of the historical development of the subject. We apologize tothose whose works we have neglected to cite and for incorrect attributions. We areindebted to many people for finding errors and misprints in the many versions ofthe material in this book and for suggesting ways to improve the exposition. Inparticular we would like to thank Ilka Agricola, Laura Barberis, Bachir Bekka, En-riqueta Rodrı́guez Carrington, Friedrich Knop, Hanspeter Kraft, Peter Landweber,and Tomasz Przebinda. Chapters of the book have been used in many courses, andthe interaction with the students was very helpful in arriving at the final version. Wethank them all for their patience, comments, and sharp eyes. During the first yearthat we were writing our previous book (1989–1990), Roger Howe gave a courseat Rutgers University on basic invariant theory. We thank him for many interestingconversations on this subject.

The first-named author is grateful for sabbatical leaves from Rutgers Universityand visiting appointments at the University of California, San Diego; the Universitéde Metz; Hong Kong University; and the National University of Singapore that weredevoted to this book. He thanks the colleagues who arranged programs in which helectured on this material, including Andrzei Hulanicki and Ewa Damek (Universityof Wrocław), Bachir Bekka and Jean Ludwig (Université de Metz), Ngai-Ming Mok(Hong Kong University), and Eng-Chye Tan and Chen-Bo Zhu (National Universityof Singapore). He also is indebted to the mathematics department of the Universityof California, Berkeley, for its hospitality during several summers of writing thebook. The second-named author would like to thank the University of California,San Diego, for sabbatical leaves that were dedicated to this project, and to thankthe National Science Foundation for summer support. We thank David Kramer for asuperb job of copyediting, and we are especially grateful to our editor Ann Kostant,who has steadfastly supported our efforts on the book from inception to publication.

New Brunswick, New Jersey Roe GoodmanSan Diego, California Nolan R. WallachFebruary 2009

Organization and Notation

Dependency Chart among Chapters and Sections

“O,” said Maggie, pouting, “I dare say I could make it out, if I’d learned what goes before,

as you have.” “But that’s what you just couldn’t, Miss Wisdom,” said Tom. “For it’s all

the harder when you know what goes before: for then you’ve got to say what Definition 3.

is and what Axiom V. is.” George Eliot, The Mill on the Floss

B. Linear and

Multilinear Algebra.......................

D. Manifolds and

Lie Groups....................................

Chap. 1 Classical Groups,

Lie Groups, and Algebraic Groups.............................................

Chap. 2 Structure of Classical Groups and Semisimple Lie Algebras.............................C. Associative and

Lie Algebras................................. Chap. 3 Highest-Weight Theory

.............................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

§4.1 Representations of Algebras§4.2 Duality for Group Representations

.................................

........................................................................................................................................................

............................................................................................................................................................................

§4.3 Group Algebras of Finite Groups§4.4 Representations of Finite Groups.............................................................................................................................................................................................................................................................................................................................................................................................................................................................

§5.1–5.4 Classical Invariant Theory.......................................................................................................

§5.5 Irreducible Representations ofClassical Groups

....................

......................................................................................

..................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................

§5.6–5.7 Applications ofInvariant Theory and Duality

Chap. 6 Spinors

Chap. 7 Character Formulas...........................

Chap. 8 Branching Laws.............................

Chaps. 9, 10 Tensor Representations

of GL(V ), O(V ), and Sp(V )

A. Algebraic

Geometry.................................................

§11.1 Structure of Algebraic Groups§11.2 Homogeneous Spaces

§11.3 Borel Subgroups.................................................................................................................................................................................................................................................

.................................................................................

§11.4 Properties of Real Forms§11.5 Gauss Decomposition

...............................................................................................................................................................................................................................................................................................

§12.1 Representations on Function Spaces..........................................................................................

...........................

...............................................................§12.2 Multiplicity-Free Spaces

..........................

§12.3 Representations on Symmetric Spaces§12.4 Isotropy Representations

of Symmetric Spaces

xix

xx Organization and Notation

Some Standard Notation

#S number of elements in set S (also denoted by Card(S) and |S|)δi j Kronecker delta (1 if i = j, 0 otherwise)

N, Z, Q nonnegative integers, integers, rational numbers

R, C, H real numbers, complex numbers, quaternions

C× multiplicative group of nonzero complex numbers

[x] greatest integer ≤ x if x is realFn n×1 column vectors with entries in field FMk,n k×n complex matrices (Mn when k = n)Mn(F) n×n matrices with entries in field FGL(n,F) invertible n×n matrices with entries from field FIn n×n identity matrix (or I when n understood)dimV dimension of a vector space V

V ∗ dual space to vector space V

〈v∗,v〉 natural duality pairing between V ∗ and VSpan(S) linear span of subset S in a vector space.

End(V ) linear transformations on vector space V

GL(V ) invertible linear transformations on vector space V

tr(A) trace of square matrix A

det(A) determinant of square matrix A

At transpose of matrix A

A∗ conjugate transpose of matrix A

diag[a1, . . . ,an] diagonal matrix⊕Vi direct sum of vector spaces Vi⊗k V k-fold tensor product of vector space V (also denoted by V⊗k)

Sk(V ) k-fold symmetric tensor product of vector space V∧k(V ) k-fold skew-symmetric tensor product of vector space VO[X ] regular functions on algebraic set X

Other notation is generally defined at its first occurrence and appears in the index ofnotation at the end of the book.

Chapter 1Lie Groups and Algebraic Groups

Abstract Hermann Weyl, in his famous book The Classical Groups, Their In-variants and Representations [164], coined the name classical groups for certainfamilies of matrix groups. In this chapter we introduce these groups and developthe basic ideas of Lie groups, Lie algebras, and linear algebraic groups. We showhow to put a Lie group structure on a closed subgroup of the general linear groupand determine the Lie algebras of the classical groups. We develop the theory ofcomplex linear algebraic groups far enough to obtain the basic results on their Liealgebras, rational representations, and Jordan–Chevalley decompositions (we deferthe deeper results about algebraic groups to Chapter 11). We show that linear al-gebraic groups are Lie groups, introduce the notion of a real form of an algebraicgroup (considered as a Lie group), and show how the classical groups introduced atthe beginning of the chapter appear as real forms of linear algebraic groups.

1.1 The Classical Groups

The classical groups are the groups of invertible linear transformations of finite-dimensional vector spaces over the real, complex, and quaternion fields, togetherwith the subgroups that preserve a volume form, a bilinear form, or a sesquilinearform (the forms being nondegenerate and symmetric or skew-symmetric).

1.1.1 General and Special Linear Groups

Let F denote either the field of real numbers R or the field of complex numbersC, and let V be a finite-dimensional vector space over F. The set of invertible lin-ear transformations from V to V will be denoted by GL(V ). This set has a groupstructure under composition of transformations, with identity element the identitytransformation I(x) = x for all x ∈ V . The group GL(V ) is the first of the classical

1R. Goodman, N.R. Wallach, Symmetry, Representations, and Invariants Graduate Texts in Mathematics 255, DOI 10.1007/978-0-387-79852-3_1, © Roe Goodman and Nolan R. Wallach 2009

,

2 1 Lie Groups and Algebraic Groups

groups. To study it in more detail, we recall some standard terminology related tolinear transformations and their matrices.

Let V and W be finite-dimensional vector spaces over F. Let {v1, . . . ,vn} and{w1, . . . ,wm} be bases for V and W , respectively. If T : V // W is a linear mapthen

T v j =m

∑i=1

ai jwi for j = 1, . . . ,n

with ai j ∈ F. The numbers ai j are called the matrix coefficients or entries of T withrespect to the two bases, and the m×n array

A =

a11 a12 · · · a1na21 a22 · · · a2n

......

. . ....

am1 am2 · · · amn

is the matrix of T with respect to the two bases. When the elements of V and W areidentified with column vectors in Fn and Fm using the given bases, then action of Tbecomes multiplication by the matrix A.

Let S : W // U be another linear transformation, with U an l-dimensional vec-tor space with basis {u1, . . . ,ul}, and let B be the matrix of S with respect to thebases {w1, . . . ,wm} and {u1, . . . ,ul}. Then the matrix of S ◦ T with respect to thebases {v1, . . . ,vn} and {u1, . . . ,ul} is given by BA, with the product being the usualproduct of matrices.

We denote the space of all n× n matrices over F by Mn(F), and we denote then× n identity matrix by I (or In if the size of the matrix needs to be indicated);it has entries δi j = 1 if i = j and 0 otherwise. Let V be an n-dimensional vectorspace over F with basis {v1, . . . ,vn}. If T : V // V is a linear map we write µ(T )for the matrix of T with respect to this basis. If T,S ∈ GL(V ) then the precedingobservations imply that µ(S ◦ T ) = µ(S)µ(T ). Furthermore, if T ∈ GL(V ) thenµ(T ◦T−1) = µ(T−1 ◦T ) = µ(Id) = I. The matrix A∈Mn(F) is said to be invertibleif there is a matrix B ∈ Mn(F) such that AB = BA = I. We note that a linear mapT : V // V is in GL(V ) if and only if its matrix µ(T ) is invertible. We also recallthat a matrix A ∈Mn(F) is invertible if and only if its determinant is nonzero.

We will use the notation GL(n,F) for the set of n× n invertible matrices withcoefficients in F. Under matrix multiplication GL(n,F) is a group with the identitymatrix as identity element. We note that if V is an n-dimensional vector space overF with basis {v1, . . . ,vn}, then the map µ : GL(V ) // GL(n,F) corresponding tothis basis is a group isomorphism. The group GL(n,F) is called the general lineargroup of rank n.

If {w1, . . . ,wn} is another basis of V , then there is a matrix g ∈ GL(n,F) suchthat

w j =n

∑i=1

gi jvi and v j =n

∑i=1

hi jwi for j = 1, . . . ,n ,

1.1 The Classical Groups 3

with [hi j] the inverse matrix to [gi j]. Suppose that T is a linear transformation fromV to V , that A = [ai j] is the matrix of T with respect to a basis {v1, . . . ,vn}, and thatB = [bi j] is the matrix of T with respect to another basis {w1, . . . ,wn}. Then

Tw j = T(∑

igi jvi

)= ∑

igi jT vi

= ∑i

gi j(∑k

akivk)

= ∑l

(∑k

∑i

hlkakigi j)

wl

for j = 1, . . . ,n. Thus B = g−1Ag is similar to the matrix A.

Special Linear Group

The special linear group SL(n,F) is the set of all elements A of Mn(F) such thatdet(A) = 1. Since det(AB) = det(A)det(B) and det(I) = 1, we see that the speciallinear group is a subgroup of GL(n,F).

We note that if V is an n-dimensional vector space over F with basis {v1, . . . ,vn}and if µ : GL(V ) // GL(n,F) is the map previously defined, then the group

µ−1(SL(n,F)) = {T ∈GL(V ) : det(µ(T )) = 1}

is independent of the choice of basis, by the change of basis formula. We denote thisgroup by SL(V ).

1.1.2 Isometry Groups of Bilinear Forms

Let V be an n-dimensional vector space over F. A bilinear map B : V ×V // Fis called a bilinear form. We denote by O(V,B) (or O(B) when V is understood)the set of all g ∈GL(V ) such that B(gv,gw) = B(v,w) for all v,w ∈V . We note thatO(V,B) is a subgroup of GL(V ); it is called the isometry group of the form B.

Let {v1, . . . ,vn} be a basis of V and let Γ ∈ Mn(F) be the matrix with Γi j =B(vi,v j). If g ∈GL(V ) has matrix A = [ai j] relative to this basis, then

B(gvi, gv j) = ∑k,l

akial jB(vk, vl) = ∑k,l

akiΓklal j .

Thus if At denotes the transposed matrix [ci j] with ci j = a ji, then the condition thatg ∈O(B) is that

Γ = AtΓ A . (1.1)

Recall that a bilinear form B is nondegenerate if B(v,w) = 0 for all w impliesthat v = 0, and likewise B(v,w) = 0 for all v implies that w = 0. In this case wehave detΓ 6= 0. Suppose B is nondegenerate. If T : V // V is linear and satisfies


B(T v,Tw) = B(v,w) for all v,w ∈ V , then det(T ) 6= 0 by formula (1.1). Hence T ∈O(B). The next two subsections will discuss the most important special cases of thisclass of groups.

Orthogonal Groups

We start by introducing the matrix groups; later we will identify these groups withisometry groups of certain classes of bilinear forms. Let O(n,F) denote the set ofall g ∈GL(n,F) such that ggt = I. That is, gt = g−1. We note that (AB)t = BtAt andif A,B ∈ GL(n,F) then (AB)−1 = B−1A−1. It is therefore obvious that O(n,F) is asubgroup of GL(n,F). This group is called the orthogonal group of n×n matricesover F. If F = R we introduce the indefinite orthogonal groups, O(p,q), with p+q =n and p,q ∈ N. Let

Ip,q =[

Ip 00 −Iq

]and define

O(p,q) = {g ∈Mn(R) : gt Ip,qg = Ip,q} .We note that O(n,0) = O(0,n) = O(n,R). Also, if

s =

0 0 · · · 1...

.... . .

...0 1 · · · 01 0 · · · 0

is the matrix with entries 1 on the skew diagonal ( j = n+1− i) and all other entries0, then s = s−1 = st and sIp,qs−1 = sIp,qs = sIp,qs =−Iq,p. Thus the map

ϕ : O(p,q) // GL(n,R)

given by ϕ(g) = sgs defines an isomorphism of O(p,q) onto O(q, p).We will now describe these groups in terms of bilinear forms.

Definition 1.1.1. Let V be a vector space over R and let M be a symmetric bilinearform on V . The form M is positive definite if M(v,v) > 0 for every v ∈V with v 6= 0.

Lemma 1.1.2. Let V be an n-dimensional vector space over F and let B be a sym-metric nondegenerate bilinear form over F.

1. If F = C then there exists a basis {v1, . . . ,vn} of V such that B(vi, v j) = δi j.2. If F = R then there exist integers p,q≥ 0 with p+q = n and a basis {v1, . . . ,vn}

of V such that B(vi, v j) = εiδi j with εi = 1 for i≤ p and εi =−1 for i > p. Fur-thermore, if we have another such basis then the corresponding integers (p,q)are the same.


Remark 1.1.3. The basis for V in part (2) is called a pseudo-orthonormal basis rel-ative to B, and the number p− q is called the signature of the form (we will alsocall the pair (p,q) the signature of B). A form is positive definite if and only if itssignature is n. In this case a pseudo-orthonormal basis is an orthonormal basis in theusual sense.

Proof. We first observe that if M is a symmetric bilinear form on V such thatM(v,v) = 0 for all v ∈ V , then M = 0. Indeed, using the symmetry and bilinear-ity we have

4M(v,w) = M(v+w, v+w)−M(v−w, v−w) = 0 for all v,w ∈V . (1.2)

We now construct a basis {w1, . . . ,wn} of V such that

B(wi, w j) = 0 for i 6= j and B(wi, wi) 6= 0

(such a basis is called an orthogonal basis with respect to B). The argument isby induction on n. Since B is nondegenerate, there exists a vector wn ∈ V withB(wn,wn) 6= 0 by (1.2). If n = 1 we are done. If n > 1, set

V ′ = {v ∈V : B(wn, v) = 0} .

For v ∈V setv′ = v− B(v, wn)

B(wn, wn)wn .

Clearly, v′ ∈V ′; hence V = V ′+Fwn. In particular, this shows that dimV ′ = n−1.We assert that the form B′ = B|V ′×V ′ is nondegenerate on V ′. Indeed, if v ∈V ′ satis-fies B(v′, w) = 0 for all w ∈V ′, then B(v′, w) = 0 for all w ∈V , since B(v′, wn) = 0.Hence v′ = 0, proving nondegeneracy of B′. We may assume by induction that thereexists a B′-orthogonal basis {w1, . . . ,wn−1} for V ′. Then it is clear that {w1, . . . ,wn}is a B-orthogonal basis for V .

If F = C let {w1, . . . ,wn} be an orthogonal basis of V with respect to B and letzi ∈ C be a choice of square root of B(wi, wi). Setting vi = (zi)−1wi, we then obtainthe desired normalization B(vi, v j) = δi j.

Now let F = R. We rearrange the indices (if necessary) so that B(wi, wi) ≥B(wi+1, wi+1) for i = 1, . . . ,n−1. Let p = 0 if B(w1, w1) < 0. Otherwise, let

p = max{i : B(wi, wi) > 0} .

Then B(wi, wi) < 0 for i > p. Take zi to be a square root of B(wi, wi) for i≤ p, andtake zi to be a square root of −B(wi, wi) for i > p. Setting vi = (zi)−1wi, we nowhave B(vi, v j) = εiδi j.

We are left with proving that the integer p is intrinsic to B. Take any basis{v1, . . . ,vn} such that B(vi, v j) = εiδi j with εi = 1 for i ≤ p and εi = −1 for i > p.Set

V+ = Span{v1, . . . ,vp} , V− = Span{vp+1, . . . ,vn} .


Then V = V+⊕V− (direct sum). Let π : V // V+ be the projection onto the firstfactor. We note that B|V+×V+ is positive definite. Let W be any subspace of V suchthat B|W×W is positive definite. Suppose that w ∈W and π(w) = 0. Then w ∈V−, soit can be written as w = ∑i>p aivi. Hence

B(w,w) = ∑i, j>p

aia j B(vi, v j) =−∑i>p

a2i ≤ 0 .

Since B|W×W has been assumed to be positive definite, it follows that w = 0. Thisimplies that π : W // V+ is injective, and hence dimW ≤ dimV+ = p. Thus pis uniquely determined as the maximum dimension of a subspace on which B ispositive definite. ut

The following result follows immediately from Lemma 1.1.2.

Proposition 1.1.4. Let B be a nondegenerate symmetric bilinear form on an n-dimensional vector space V over F.

1. Let F = C. If {v1, . . . ,vn} is an orthonormal basis for V with respect to B, thenµ : O(V,B) // O(n,F) defines a group isomorphism.

2. Let F = R. If B has signature (p,n− p) and {v1, . . . ,vn} is a pseudo-orthonormalbasis of V , then µ : O(V,B) // O(p,n− p) is a group isomorphism.

Here µ(g), for g ∈GL(V ), is the matrix of g with respect to the given basis.The special orthogonal group over F is the subgroup

SO(n,F) = O(n,F)∩SL(n,F)

of O(n,F). The indefinite special orthogonal groups are the groups

SO(p,q) = O(p,q)∩SL(p+q,R) .

Symplectic Group

We set J =[

0 I−I 0

]with I the n×n identity matrix. The symplectic group of rank n

over F is defined to be

Sp(n,F) = {g ∈M2n(F) : gtJg = J}.

As in the case of the orthogonal groups one sees without difficulty that Sp(n,F) is asubgroup of GL(2n,F).

We will now look at the coordinate-free version of these groups. A bilinear formB is called skew-symmetric if B(v,w) =−B(w,v). If B is skew-symmetric and non-degenerate, then m = dimV must be even, since the matrix of B relative to any basisfor V is skew-symmetric and has nonzero determinant.


Lemma 1.1.5. Let V be a 2n-dimensional vector space over F and let B be a nonde-generate, skew-symmetric bilinear form on V . Then there exists a basis {v1, . . . ,v2n}for V such that the matrix [B(vi,v j)] equals J (call such a basis a B-symplectic ba-sis).

Proof. Let v be a nonzero element of V . Since B is nondegenerate, there exists w∈Vwith B(v,w) 6= 0. Replacing w with B(v,w)−1w, we may assume that B(v,w) = 1.Let

W = {x ∈V : B(v,x) = 0 and B(w,x) = 0} .For x ∈V we set x′ = x−B(v,x)w−B(x,w)v. Then

B(v,x′) = B(v,x)−B(v,x)B(v,w)−B(w,x)B(v,v) = 0 ,

since B(v,w) = 1 and B(v,v) = 0 (by skew symmetry of B). Similarly,

B(w,x′) = B(w,x)−B(v,x)B(w,w)+B(w,x)B(w,v) = 0 ,

since B(w,v) = −1 and B(w,w) = 0. Thus V = U ⊕W , where U is the span of vand w. It is easily verified that B|U×U is nondegenerate, and so U ∩W = {0}. Thisimplies that dimW = m− 2. We leave to the reader to check that B|W×W also isnondegenerate.

Set vn = v and v2n = w with v,w as above. Since B|W×W is nondegenerate, byinduction there exists a B-symplectic basis {w1, . . . ,w2n−2} of W . Set vi = wi andvn+1−i = wn−i for i≤ n−1. Then {v1, . . . ,v2n} is a B-symplectic basis for V . ut

The following result follows immediately from Lemma 1.1.5.

Proposition 1.1.6. Let V be a 2n-dimensional vector space over F and let B be anondegenerate skew-symmetric bilinear form on V . Fix a B-symplectic basis of Vand let µ(g), for g ∈ GL(V ), be the matrix of g with respect to this basis. Thenµ : O(V,B) // Sp(n,F) is a group isomorphism.

1.1.3 Unitary Groups

Another family of classical subgroups of GL(n,C) consists of the unitary groupsand special unitary groups for definite and indefinite Hermitian forms. If A∈Mn(C)we will use the standard notation A∗= A t for its adjoint matrix, where A is the matrixobtained from A by complex conjugating all of the entries. The unitary group of rankn is the group

U(n) = {g ∈Mn(C) : g∗g = I} .The special unitary group is SU(n) = U(n)∩SL(n,C). Let the matrix Ip,q be as inSection 1.1.2. We define the indefinite unitary group of signature (p,q) to be

U(p,q) = {g ∈Mn(C) : g∗Ip,qg = Ip,q} .


The special indefinite unitary group of signature (p,q) is SU(p,q) = U(p,q) ∩SL(n,C).

We will now obtain a coordinate-free description of these groups. Let V be ann-dimensional vector space over C. An R bilinear map B : V ×V // C (where weview V as a vector space over R) is said to be a Hermitian form if it satisfies

1. B(av,w) = aB(v,w) for all a ∈ C and all v,w ∈V .2. B(w,v) = B(v,w) for all v,w ∈V .By the second condition, we see that a Hermitian form is nondegenerate providedB(v,w) = 0 for all w ∈V implies that v = 0. The form is said to be positive definiteif B(v,v) > 0 for all v ∈ V with v 6= 0. (Note that if M is a Hermitian form, thenM(v,v) ∈ R for all v ∈ V .) We define U(V,B) (also denoted by U(B) when V isunderstood) to be the group of all elements g∈GL(V ) such that B(gv,gw) = B(v,w)for all v,w ∈V . We call U(B) the unitary group of B.

Lemma 1.1.7. Let V be an n-dimensional vector space over C and let B be a non-degenerate Hermitian form on V . Then there exist an integer p, with n≥ p≥ 0, anda basis {v1, . . . ,vn} of V such that B(vi,v j) = εiδi j, with εi = 1 for i≤ p and εi =−1for i > p. The number p depends only on B and not on the choice of basis.

The proof of Lemma 1.1.7 is almost identical to that of Lemma 1.1.2 and will beleft as an exercise.

If V is an n-dimensional vector space over C and B is a nondegenerate Hermitianform on V , then a basis as in Lemma 1.1.7 will be called a pseudo-orthonormal basis(if p = n then it is an orthonormal basis in the usual sense). The pair (p,n− p) willbe called the signature of B. The following result is proved in exactly the same wayas the corresponding result for orthogonal groups.

Proposition 1.1.8. Let V be a finite-dimensional vector space over C and let B be anondegenerate Hermitian form on V of signature (p,q). Fix a pseudo-orthonormalbasis of V relative to B and let µ(g), for g ∈GL(V ), be the matrix of g with respectto this basis. Then µ : U(V,B) // U(p,q) is a group isomorphism.

1.1.4 Quaternionic Groups

We recall some basic properties of the quaternions. Consider the four-dimensionalreal vector space H consisting of the 2×2 complex matrices

w =[

x −yy x

]with x,y ∈ C . (1.3)

One checks directly that H is closed under multiplication in M2(C). If w ∈ H thenw∗ ∈H and

w∗w = ww∗ = (|x|2 + |y|2)I


(where w∗ denotes the conjugate-transpose matrix). Hence every nonzero elementof H is invertible. Thus H is a division algebra (or skew field) over R. This divisionalgebra is a realization of the quaternions.

The more usual way of introducing the quaternions is to consider the vector spaceH over R with basis {1, i, j, k}. Define a multiplication so that 1 is the identity and

i2 = j2 = k2 =−1 ,

ij =−ji = k , ki =−ik = j , jk =−kj = i ;then extend the multiplication to H by linearity relative to real scalars. To obtain anisomorphism between this version of H and the 2×2 complex matrix version, take

1 = I , i =[

i 00 −i

], j =

[0 1−1 0

], k =

[0 ii 0

],

where i is a fixed choice of√−1. The conjugation w 7→ w∗ satisfies (uv)∗ = v∗u∗.

In terms of real components, (a+bi+cj+dk)∗ = a−bi−cj−dk for a,b,c,d ∈R.It is useful to write quaternions in complex form as x + jy with x,y ∈ C; however,note that the conjugation is then given as

(x+ jy)∗ = x+ yj = x− jy .

On the 4n-dimensional real vector space Hn we define multiplication by a ∈ Hon the right:

(u1, . . . ,un) ·a = (u1a, . . . ,una) .We note that u · 1 = u and u · (ab) = (u · a) · b. We can therefore think of Hn as avector space over H. Viewing elements of Hn as n× 1 column vectors, we defineAu for u ∈ Hn and A ∈ Mn(H) by matrix multiplication. Then A(u · a) = (Au) · afor a ∈ H; hence A defines a quaternionic linear map. Here matrix multiplicationis defined as usual, but one must be careful about the order of multiplication of theentries.

We can make Hn into a 2n-dimensional vector space over C in many ways; forexample, we can embed C into H as any of the subfields

R1+Ri , R1+Rj , R1+Rk . (1.4)

Using the first of these embeddings, we write z = x + jy ∈ Hn with x,y ∈ Cn, andlikewise C = A+ jB ∈Mn(H) with A,B ∈Mn(C). The maps

z 7→[

xy

]and C 7→

[A −BB A

]identify Hn with C2n and Mn(H) with the real subalgebra of M2n(C) consisting ofmatrices T such that


JT = T J , where J =[

0 I−I 0

]. (1.5)

We define GL(n,H) to be the group of all invertible n×n matrices over H. ThenGL(n,H) acts on Hn by complex linear transformations relative to each of the com-plex structures (1.4). If we use the embedding of Mn(H) into M2n(C) just described,then from (1.5) we see that GL(n,H) = {g ∈GL(2n,C) : Jg = gJ}.

Quaternionic Special Linear Group

We leave it to the reader to prove that the determinant of A∈GL(n,H) as a complexlinear transformation with respect to any of the complex structures (1.4) is the same.We can thus define SL(n,H) to be the elements of determinant one in GL(n,H)with respect to any of these complex structures. This group is usually denoted bySU∗(2n).

The Quaternionic Unitary Groups

For X = [xi j] ∈ Mn(H) we define X∗ = [x∗ji] (here we take the quaternionic matrixentries xi j ∈M2(C) given by (1.3)). Let the diagonal matrix Ip,q (with p+q = n) beas in Section 1.1.2. The indefinite quaternionic unitary groups are the groups

Sp(p,q) = {g ∈GL(p+q,H) : g∗Ip,qg = Ip,q} .

We leave it to the reader to prove that this set is a subgroup of GL(p+q,H).The group Sp(p,q) is the isometry group of the nondegenerate quaternionic Her-

mitian formB(w,z) = w∗Ip,qz, for w,z ∈Hn . (1.6)

(Note that this form satisfies B(w,z) = B(z,w)∗ and B(wα,zβ ) = α∗B(w,z)β forα,β ∈ H.) If we write w = u + jv and z = x + jy with u,v,x,y ∈ Cn, and set Kp,q =diag[Ip,q Ip,q] ∈M2n(R), then

B(w,z) =[

u∗ v∗]

Kp,q

[xy

]+ j[

ut vt]

Kp,q

[−yx

].

Thus the elements of Sp(p,q), viewed as linear transformations of C2n, preserveboth a Hermitian form of signature (2p,2q) and a nondegenerate skew-symmetricform.


The Group SO∗(2n)

Let J be the 2n× 2n skew-symmetric matrix from Section 1.1.2. Since J2 = −I2n(the 2n×2n identity matrix), the map of GL(2n,C) to itself given by θ(g) =−JgJdefines an automorphism whose square is the identity. Our last family of classicalgroups is

SO∗(2n) = {g ∈ SO(2n,C) : θ(g) = g} .We identify C2n with Hn as a vector space over C by the map

[ab]7→ a+ jb, where

a,b ∈ Cn. The group SO∗(2n) then becomes the isometry group of the nondegener-ate quaternionic skew-Hermitian form

C(x,y) = x∗jy, for x,y ∈Hn . (1.7)

This form satisfies C(x,y) =−C(y,x)∗ and C(xα,yβ ) = α∗C(x,y)β for α,β ∈H.We have now completed the list of the classical groups associated with R, C, and

H. We will return to this list at the end of the chapter when we consider real forms ofcomplex algebraic groups. Later we will define covering groups; any group coveringone of the groups on this list—for example, a spin group in Chapter 7—will also becalled a classical group.

1.1.5 Exercises

In these exercises F denotes either R or C. See Appendix B.2 for notation andproperties of tensor and exterior products of vector spaces.

1. Let {v1, . . . ,vn} and {w1, . . . ,wn} be bases for an F vector space V . Suppose alinear map T : V // V has matrices A and B, respectively, relative to thesebases. Show that detA = detB.

2. Determine the signature of the form B(x,y) = ∑ni=1 xiyn+1−i on Rn.3. Let V be a vector space over F and let B be a skew-symmetric or symmetric

nondegenerate bilinear form on V . Assume that W is a subspace of V on which Brestricts to a nondegenerate form. Prove that the restriction of B to the subspaceW⊥ = {v ∈V : B(v,w) = 0 for all w ∈W} is nondegenerate.

4. Let V denote the vector space of symmetric 2× 2 matrices over F. If x,y ∈ Vdefine B(x,y) = det(x+ y)−det(x)−det(y).(a) Show that B is nondegenerate, and that if F = R then the signature of the formB is (1,2).(b) If g ∈ SL(2,F) define ϕ(g) ∈GL(V ) by ϕ(g)(v) = gvgt . Show that the mapϕ : SL(2,F) // SO(V,B) is a group homomorphism with kernel {±I}.

5. The purpose of this exercise is to prove Lemma 1.1.7 by the method of proof ofLemma 1.1.2.


(a) Prove that if M is a Hermitian form such that M(v,v) = 0 for all v then M = 0.(HINT: Show that M(v + sw,v + sw) = sM(w,v) + sM(v,w) for all s ∈ C; thensubstitute values for s to see that M(v,w) = 0.)(b) Use the result of part (a) to complete the proof of Lemma 1.1.7.(HINT: Note that M(v,v) ∈ R since M is Hermitian.)

6. Let V be a 2n-dimensional vector space over F. Consider the space W =∧nV .

Fix a basis ω of the one-dimensional vector space∧2nV . Consider the bilinear

form B(u,v) on W defined by u∧ v = B(u,v)ω .(a) Show that B is nondegenerate.(b) Show that B is skew-symmetric if n is odd and symmetric if n is even.(c) Determine the signature of B when n is even and F = R,

7. (Notation of the previous exercise) Let V = F4 with basis {e1,e2,e3,e4} and letω = e1∧e2∧e3∧e4. Define ϕ(g)(u∧v) = gu∧gv for g∈ SL(4,F) and u,v∈ F4.Show that ϕ : SL(4,F) // SO(

∧2F4,B) is a group homomorphism with kernel{±I}. (HINT: Use Jordan canonical form to determine the kernel.)

8. (Notation of the previous exercise) Let ψ be the restriction of ϕ to Sp(2,F). Letν = e1∧ e3 + e2∧ e4.(a) Show that ψ(g)ν = ν and B(ν ,ν) = −2. (HINT: The map ei ∧ e j 7→ ei j −e ji is a linear isomorphism between

∧2 F4 and the subspace of skew-symmetricmatrices in M4(F). Show that this map takes ν to J and ϕ(g) to the transformationA 7→ gAgt .)(b) Let W = {w ∈∧2F4 : B(ν ,w) = 0}. Show that B|W×W is nondegenerate andhas signature (3,2) when F = R.(c) Set ρ(g) = ψ(g)|W . Show that ρ is a group homomorphism from Sp(2,F) toSO(W,B|W×W ) with kernel {±1}. (HINT: Use the previous exercise to determinethe kernel.)

9. Let V = M2(F). For x,y ∈V define B(x,y) = det(x+ y)−det(x)−det(y).(a) Show that B is a symmetric nondegenerate form on V , and calculate the sig-nature of B when F = R.(b) Let G = SL(2,F)×SL(2,F) and define ϕ : G // GL(V ) by ϕ(a,b)v = axbtfor a,b∈ SL(2,F) and v∈V . Show that ϕ is a group homomorphism and ϕ(G)⊂SO(V,B). Determine Ker(ϕ). (HINT: Use Jordan canonical form to determinethe kernel.)

10. Identify Hn with C2n as a vector space over C by the map a + jb 7→[a

b], where

a,b ∈ Cn. Let T = A+ jB ∈Mn(H) with A,B ∈Mn(C).(a) Show that left multiplication by T on Hn corresponds to multiplication by thematrix

[A −B̄B Ā

]∈M2n(C) on C2n.

(b) Show that multiplication by i on Mn(H) becomes the transformation[A −B̄B −Ā

]7→[

iA −iB̄−iB −iĀ

].

11. Use the identification of Hn with C2n in the previous exercise to view the formB(x,y) in equation (1.6) as an H-valued function on C2n×C2n.

1.2 The Classical Lie Algebras 13

(a) Show that B(x,y) = B0(x,y)+ jB1(x,y), where B0 is a C-Hermitian form onC2n of signature (2p,2q) and B1 is a nondegenerate skew-symmetric C-bilinearform on C2n.(b) Use part (a) to prove that Sp(p,q) = Sp(C2n,B1)∩U(C2n,B0).

12. Use the identification of Hn with C2n in the previous exercise to view the formC(x,y) from equation (1.7) as an H-valued function on C2n×C2n.(a) Show that C(x,y) = C0(x,y) + jxty for x,y ∈ C2n, where C0(x,y) is a C-Hermitian form on C2n of signature (n,n).(b) Use the result of part (a) to prove that SO∗(2n) = SO(2n,C)∩U(C2n,C0).

13. Why can’t we just define SL(n,H) by taking all g∈GL(n,H) such that the usualformula for the determinant of g yields 1?

14. Consider the three embeddings of C in H given by the subfields (1.4). These givethree ways of writing X ∈ Mn(H) as a 2n× 2n matrix over C. Show that thesethree matrices have the same determinant.

1.2 The Classical Lie Algebras

Let V be a vector space over F. Let End(V ) denote the algebra (under composition)of F-linear maps of V to V . If X ,Y ∈ End(V ) then we set [X ,Y ] = XY −Y X . Thisdefines a new product on End(V ) that satisfies two properties:

(1) [X ,Y ] =−[Y,X ] for all X ,Y (skew symmetry) .(2) [X , [Y,Z]] = [[X ,Y ],Z]+ [Y, [X ,Z]] for all X ,Y,Z (Jacobi identity) .

Definition 1.2.1. A vector space g over F together with a bilinear map X ,Y 7→ [X ,Y ]of g×g to g is said to be a Lie algebra if conditions (1) and (2) are satisfied.

In particular, End(V ) is a Lie algebra under the binary operation [X ,Y ] =XY −Y X . Condition (2) is a substitute for the associative rule for multiplication;it says that for fixed X , the linear transformation Y 7→ [X ,Y ] is a derivation of the(nonassociative) algebra (g, [· , ·]).

If g is a Lie algebra and if h is a subspace such that X ,Y ∈ h implies that [X ,Y ] inh, then h is a Lie algebra under the restriction of [· , ·]. We will call h a Lie subalgebraof g (or subalgebra, when the Lie algebra context is clear).

Suppose that g and h are Lie algebras over F. A Lie algebra homomorphism ofg to h is an F-linear map T : g // h such that T [X ,Y ] = [T X ,TY ] for all X ,Y ∈ g.A Lie algebra homomorphism is an isomorphism if it is bijective.

1.2.1 General and Special Linear Lie Algebras

If V is a vector space over F, we write gl(V ) for End(V ) looked upon as a Liealgebra under [X ,Y ] = XY −Y X . We write gl(n,F) to denote Mn(F) as a Lie algebra


under the matrix commutator bracket. If dimV = n and we fix a basis for V , thenthe correspondence between linear transformations and their matrices gives a Liealgebra isomorphism gl(V )∼= gl(n,R). These Lie algebras will be called the generallinear Lie algebras.

If A = [ai j]∈Mn(F) th

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