Rolf Berndt Representations of Linear Groups Rolf Berndt Representations of Linear Groups An Introduction Based on Examples from Physics and Number Theory Bibliografische information published by Die Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutschen Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.d-nb.de>. Prof. Dr. Rolf Berndt Department of Mathematics University of Hamburg Bundesstraße 55 D-20146 Hamburg Germany [email protected] Mathematics Subject Classification 20G05, 22E45, 20C25, 20C35, 11F70 First edition, July 2007 All rights reserved © Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden 2007 Editorial Office: Ulrike Schmickler-Hirzebruch | Susanne Jahnel Vieweg is a company in the specialist publishing group Springer Science+Business Media. www.vieweg.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Cover design: Ulrike Weigel, www.CorporateDesignGroup.de Printing and binding: MercedesDruck, Berlin Printed on acid-free paper Printed in Germany ISBN 978-3-8348-0319-1 Preface There are already many good books on representation theory for all kinds of groups. Two of the best (in this author’s opinion) are the one by A.W. Knapp: “Representation Theory for Semisimple Groups. An Overview based on Examples” [Kn1] and by G.W. Mackey: “Induced Representations in Physics, Probability and Number Theory” [Ma1]. The title of this text is a mixture of both these titles, and our text is meant as a very elementary introduction to these and, moreover, to the whole topic of group represen- tations, even infinite-dimensional ones. As is evident from the work of Bargmann [Ba], Weyl [Wey] and Wigner [Wi], group representations are fundamental for the theory of atomic spectra and elementary physics. But representation theory has proven to be an inavoidable ingredient in other fields as well, particularly in number theory, as in the theory of theta functions, automorphic forms, Galois representations and, finally, the Langlands program. Hence, we present an approach as elementary as possible, having in particular these applications in mind. This book is written as a summary of several courses given in Hamburg for students of Mathematics and Physics from the fifth semester on. Thus, some knowledge of linear and multilinear algebra, calculus and analysis in several variables is taken for granted. Assuming these prerequisites, several groups of particular interest for the applications in physics and number theory are presented and discussed, including the symmetric group Sn as the leading example for a finite group, the groups SO(2), SO(3), SU(2), and SU(3) as examples of compact groups, the Heisenberg groups and SL(2, R), SL(2, C), resp. the Lorentz group SO(3, 1) as examples for noncompact groups, and the Euclidean groups E(n) = SO(n) Rn and the Poincar´egroup P = SO(3, 1)+ R4 as examples for semidi- rect products. This text would not have been possible without the assistance of my students and colleagues; it is a pleasure for me to thank them all. In particular, D. Bahns, S. B¨ocherer, O. v. Grudzinski, M. Hohmann, H. Knorr, J. Michaliˇcek,H. M¨uller,B. Richter, R. Schmidt, and Chr. Schweigert helped in many ways, from giving valuable hints to indicating several mistakes. Part of the material was treated in a joint seminar with Peter Slodowy. I hope that a little bit of his way of thinking is still felt in this text and that it is apt to participate in keeping alive his memory. Finally, I am grateful to U. Schmickler-Hirzebruch and S. Jahnel from the Vieweg Verlag for encouragement and good advice. Contents Introduction ix 0 Prologue: Some Groups and their Actions 1 0.1 Several Matrix Groups ............................. 1 0.2 Group Actions ................................. 3 0.3 The Symmetric Group ............................. 5 1 Basic Algebraic Concepts 7 1.1 Linear Representations ............................. 7 1.2 Equivalent Representations .......................... 9 1.3 First Examples ................................. 10 1.4 Basic Construction Principles ......................... 14 1.4.1 Sum of Representations ........................ 14 1.4.2 Tensor Product of Representations .................. 14 1.4.3 The Contragredient Representation .................. 15 1.4.4 The Factor Representation ...................... 16 1.5 Decompositions ................................. 16 1.6 Characters .................................... 21 2 Representations of Finite Groups 23 2.1 Characters as Orthonormal Systems ..................... 23 2.2 The Regular Representation .......................... 27 2.3 Characters as Orthonormal Bases ...................... 28 3 Continuous Representations 31 3.1 Topological and Linear Groups ........................ 31 3.2 The Continuity Condition ........................... 33 3.3 Invariant Measures ............................... 38 3.4 Examples .................................... 40 4 Representations of Compact Groups 43 4.1 Basic Facts ................................... 43 4.2 The Example G = SU(2) ............................ 48 4.3 The Example G = SO(3) ........................... 52 5 Representations of Abelian Groups 59 5.1 Characters and the Pontrjagin Dual ..................... 59 5.2 Continuous Decompositions .......................... 60 viii CONTENTS 6 The Infinitesimal Method 63 6.1 Lie Algebras and their Representations .................... 63 6.2 The Lie Algebra of a Linear Group ...................... 67 6.3 Derived Representations ............................ 70 6.4 Unitarily Integrable Representations of sl(2, R)............... 73 6.5 The Examples su(2) and heis(R) ....................... 82 6.6 Some Structure Theory ............................ 84 6.6.1 Specifications of Groups and Lie Algebras .............. 85 6.6.2 Structure Theory for Complex Semisimple Lie Algebras ...... 89 6.6.3 Structure Theory for Compact Real Lie Algebras .......... 93 6.6.4 Structure Theory for Noncompact Real Lie Algebras ........ 95 6.6.5 Representations of Highest Weight .................. 97 6.7 The Example su(3)...............................104 7 Induced Representations 117 7.1 The Principle of Induction . ........................117 7.1.1 Preliminary Approach .........................118 7.1.2 Mackey’s Approach ...........................120 7.1.3 Final Approach .............................125 7.1.4 Some Questions and two Easy Examples ...............126 7.2 Unitary Representations of SL(2, R) .....................130 7.3 Unitary Representations of SL(2,C) and of the Lorentz Group .......143 7.4 Unitary Representations of Semidirect Products ...............147 7.5 Unitary Representations of the Poincar´eGroup ...............154 7.6 Induced Representations and Vector Bundles ................161 8 Geometric Quantization and the Orbit Method 173 8.1 The Hamiltonian Formalism and its Quantization ..............173 8.2 Coadjoint Orbits and Representations ....................178 8.2.1 Prequantization .............................178 8.2.2 Example: Construction of Line Bundles over M = P1(C) .....181 8.2.3 Quantization ..............................184 8.2.4 Coadjoint Orbits and Hamiltonian G-spaces .............186 8.2.5 Construction of an Irreducible Unitary Representation by an Orbit 196 8.3 The Examples SU(2) and SL(2, R) ......................197 8.4 The Example Heis(R) .............................202 8.5 Some Hints Concerning the Jacobi Group ..................209 9 Epilogue: Outlook to Number Theory 215 9.1 Theta Functions and the Heisenberg Group .................216 9.2 Modular Forms and SL(2,R) .........................221 9.3 Theta Functions and the Jacobi Group ....................236 9.4 Hecke’s Theory of L−Functions Associated to Modular Forms .......239 9.5 Elements of Algebraic Number Theory and Hecke L-Functions ......246 9.6 Arithmetic L-Functions ............................250 9.7 Summary and Final Reflections ........................256 Bibliography 261 Index 266 Introduction In this book, the groups enumerated in the Preface are introduced and treated as matrix groups to avoid as much as possible the machinery of manifolds, Lie groups and bundles (though some of it soon creeps in through the backdoor as the theory is further devel- oped). Parallel to information about the structure of our groups we shall introduce and develop elements of the representation theory necessary to classify the unitary represen- tations and to construct concrete models for these representations. As the main tool for the classification we use the infinitesimal method linearizing the representations of a group by studying those of the Lie algebra of the group. And as the main tools for the construction of models for the representations we use – tensor products of the natural representation, – representations given by smooth functions (in particular polynomials) living on a space provided with an action of the group, and – the machinery of induced representations. Moreover, because of the growing importance in physics and the success in deriving branching relations, the procedure of geometric quantization and the orbit method, de- veloped and propagated by Kirillov, Kostant, Duflo and many others shall be explained via its application to some of the examples above. Besides the sources already mentioned, the author was largely influenced by the now classical book of Kirillov: