An Introduction to the Representation Theory of Groups Emmanuel Kowalski Graduate Studies in Mathematics Volume 155 American Mathematical Society An Introduction to the Representation Theory of Groups https://doi.org/10.1090//gsm/155 An Introduction to the Representation Theory of Groups Emmanuel Kowalski Graduate Studies in Mathematics Volume 155 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 20-01, 20Cxx, 22A25. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-155 Library of Congress Cataloging-in-Publication Data Kowalski, Emmanuel, 1969– An introduction to the representation theory of groups / Emmanuel Kowalski. pages cm. — (Graduate studies in mathematics ; volume 155) Includes bibliographical references and index. ISBN 978-1-4704-0966-1 (alk. paper) 1. Lie groups. 2. Representations of groups. 3. Group algebras. I. Title. QA387.K69 2014 515.7223—dc23 2014012974 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents Chapter 1. Introduction and motivation 1 §1.1. Presentation 3 §1.2. Four motivating statements 4 §1.3. Prerequisites and notation 8 Chapter 2. The language of representation theory 13 §2.1. Basic language 13 §2.2. Formalism: changing the space 21 §2.3. Formalism: changing the group 42 §2.4. Formalism: changing the field 65 §2.5. Matrix representations 68 §2.6. Examples 70 §2.7. Some general results 80 §2.8. Some Clifford theory 121 §2.9. Conclusion 124 Chapter 3. Variants 127 §3.1. Representations of algebras 127 §3.2. Representations of Lie algebras 132 §3.3. Topological groups 139 §3.4. Unitary representations 145 Chapter 4. Linear representations of finite groups 159 §4.1. Maschke’s Theorem 159 v vi Contents §4.2. Applications of Maschke’s Theorem 163 §4.3. Decomposition of representations 169 §4.4. Harmonic analysis on finite groups 190 §4.5. Finite abelian groups 200 §4.6. The character table 208 §4.7. Applications 240 §4.8. Further topics 262 Chapter 5. Abstract representation theory of compact groups 269 §5.1. An example: the circle group 269 §5.2. The Haar measure and the regular representation of a locally compact group 272 §5.3. The analogue of the group algebra 288 §5.4. The Peter–Weyl Theorem 294 §5.5. Characters and matrix coefficients for compact groups 304 §5.6. Some first examples 310 Chapter 6. Applications of representations of compact groups 319 §6.1. Compact Lie groups are matrix groups 319 §6.2. The Frobenius–Schur indicator 324 §6.3. The Larsen alternative 332 §6.4. The hydrogen atom 344 Chapter 7. Other groups: a few examples 355 §7.1. Algebraic groups 355 §7.2. Locally compact groups: general remarks 369 §7.3. Locally compact abelian groups 371 §7.4. A non-abelian example: SL2pRq 376 Appendix A. Some useful facts 409 §A.1. Algebraic integers 409 §A.2. The spectral theorem 414 §A.3. The Stone–Weierstrass Theorem 420 Bibliography 421 Index 425 Bibliography [1] J. Baez and J. Huerta: The algebra of Grand Unified Theories, Bulletin of the American Mathematical Society 47 (2010), 483–552. [2]B.Bekka,P.delaHarpe,andA.Valette: Kazhdan’s Property pT q,NewMath. Monographs 11, Cambridge University Press, 2008. [3] J. Bernstein and S. Gelbart (eds): An introduction to the Langlands program, Birkha¨user, 2003. [4]J.BernsteinandA.Reznikov: Analytic continuation of representations and es- timates of automorphic forms, Annals of Math. 150 (1999), 329–352. [5] N. Berry, A. Dubickas, N. Elkies, B. Poonen, and C. J. Smyth: The conjugate dimension of algebraic numbers, Quart. J. Math. 55 (2004), 237–252. [6]E.D.Bolker: The spinor spanner, American Math. Monthly 80 (1973), 977–984. [7] A. 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