Representation Theory and Noncommutative Harmonic Analysis I: Fundamental Concepts. Representations of Virasoro and Affine Algebras

240 Pages·1994·7.361 MB·English

by  A. A. Kirillov (auth.), A. A. Kirillov (eds.)

Encyclopaedia of
Mathematical Sciences
Volume 22
Editor-in-Chief: R. V. Gamkrelidze
A.A. Kirillov (Ed.)
Representation Theory
and Noncommutative
Harmonic Analysis I
Fundamental Concepts. Representations of Virasoro
and Affine Algebras
With 11 Figures
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
A.A. Agrachev, A.A. Gonchar, E. F. Mishchenko,
N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundarnental'nye napravleniya, Vol. 22, Teoriya predstavlenij i nekommutativnyj
garmonicheskij analiz 1
Publisher VINITI, Moscow 1988
Mathematics Subject Classification (1991):
16Gxx, 17B68
ISBN 978-3-642-05740-3
Library of Congress Cataloging-in-Publication Data
Teoriia predstavlenii i nekommutativnyl garmonicheskil analiz I.
English. Representation theory and noncommutative harmonic analysis I:
fundamental concepts, representations of Virasoro and affine algebras I A. A. Kirillov (ed.) p. em. -
(Encyclopaedia of mathematical sciences; v. 22)
Includes bibliographical references and indexes.
ISBN 978-3-642-05740-3 ISBN 978-3-662-03002-8 (eBook)
DOI 10.1007/978-3-662-03002-8
I. Representations of groups. 2. Kac-Moody algebras. 3. Harmonic analysis.
I. Kirillov, A. A. (Aieksandr Aleksandrovich), 1936-. II. Title. III. Series.
QA176.T4613 1994 512'.55-dc20 94-11597
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© Springer-Verlag Berlin Heidelberg 1994
Originally published by Springer-Verlag Berlin Heidelberg New York in 1994
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List of Editors, Authors and Translators
Editor-in-Chief
R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information {VINITI),
ul. Usievicha 20a, 125219 Moscow
Consulting Editor
A. A. Kirillov, Moscow University, Mehmat, 117234 Moscow, Russia
Authors
A. A. Kirillov, Moscow University, Mehmat, 117234 Moscow, Russia
Yu. A. Neretin, Moscow Institute of Electronic Engineering, MIEM,
Bol'shoj Vuzovskij Per. 3/12, 109028 Moscow, Russia
Translator
V. Soucek, Charles University, Mathematical Institute, Sokolovska 83,
18600 Prague, Czech Republic
Contents
I. Introduction to the Theory of Representations
and Noncommutative Harmonic Analysis
A. A. Kirillov
1
II. Representations of Virasoro and Affine Lie Algebra!
Yu. A. Neretin
157
Author Index
227
Subject Index
229
I. Introduction to the Theory
of Representations
and Noncommutative Harmonic
Analysis
A. A. Kirillov
Translated from the Russian
by V. Soucek
Contents
Chapter 1. A Historical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§1 . Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§2. Finite-Dimensional Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§3. Infinite-Dimensional Representations . . . . . . . . . . . . . . . . . . . . . . . . . 7
§4. The General Theory of Infinite-Dimensional Representations . . . . . 9
§5. Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
§6. Representations of Semisimple Groups . . . . . . . . . . . . . . . . . . . . . . . . 10
§7. The Method of Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§8. Infinite-Dimensional Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
§9. Representation of Lie Supergroups and Superalgebras . . . . . . . . . . . 14
Chapter 2. Basic Notions of the Theory of Representations . . . . . . . . . . 15
§1. Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1. Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2. The Category of G-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3. Actions of Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 18
§2. Linear Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1. Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 A. A. Kirillov
2.2. The Category of Linear Representations . . . . . . . . . . . . . . . . . . 21
2.3. Projective Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
§3. Noncommutative Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1. The Classification of Representations . . . . . . . . . . . . . . . . . . . . . 25
3.2. The Computation of the Spectrum of a Representation . . . . . 25
3.3. The Functors Res and Ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4. The Fourier Transform on a Group . . . . . . . . . . . . . . . . . . . . . . 26
3.5. Special Functions and Representation Theory . . . . . . . . . . . . . 27
3.6. The Computation of Generalized and Infinitesimal
Characters of Representations of Lie Groups . . . . . . . . . . . . . . 27
Chapter 3. Representations of Finite Groups . . . . . . . . . . . . . . . . . . . . . . 28
§1. The General Theory of Complex Finite-Dimensional
Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.1. The Formulation of Basic Results . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2. Schur's Lemma and Its Consequences . . . . . . . . . . . . . . . . . . . . 29
§2. The Theory of Characters and Group Algebras . . . . . . . . . . . . . . . . 32
2.1. Basic Properties of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2. The Group Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3. The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
§3. The Decomposition of Representations . . . . . . . . . . . . . . . . . . . . . . . . 36
§4. The Connection Between Representations of a Group
and Its Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1. The Functors Res and Ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2. Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3. Big and Spherical Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
§5. The Representation Ring. Operations on Representations . . . . . . . 43
5.1. Virtual Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2. Operations on Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 44
§6. Representations over Other Fields and Rings . . . . . . . . . . . . . . . . . . 46
6.1. Basic Definitions and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2. Real Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.3. Integer and Modular Representations . . . . . . . . . . . . . . . . . . . . 49
§7. Projective Representations of Finite Groups . . . . . . . . . . . . . . . . . . . 49
§8. Representations of the Symmetric Group . . . . . . . . . . . . . . . . . . . . . . 50
8.1. Notation and Subsidiary Constructions . . . . . . . . . . . . . . . . . . . 50
8.2. Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.3. Examples of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.4. Branching Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.5. The Ring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4. Representations of Compact Groups . . . . . . . . . . . . . . . . . . . . 57
§1. Invariant Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.1. The Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
I. Theory of Representations and Noncommutative Harmonic Analysis 3
1.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.3. Integration of Vector and Operator Valued Functions . . . . . . 60
§2. General Properties of Representations . . . . . . . . . . . . . . . . . . . . . . . . 61
2.1. The Formulation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3. Group Algebras and the Fourier Transform . . . . . . . . . . . . . . . 63
2.4. The Decomposition of Representations . . . . . . . . . . . . . . . . . . . 65
§3. Representations of Groups SU(2) and S0(3) . . . . . . . . . . . . . . . . . . . 66
3.1. The Group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2. The Group S0(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3. Harmonic Analysis on the Two-Dimensional Sphere . . . . . . . . 71
Chapter 5. Finite-Dimensional Representations of a Lie Group . . . . . . . 73
§1. Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
§2. Representations of Solvable Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 78
§3. The Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
§4. Laplace (Casimir) Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
§5. Representations of the Group SU(2) (Infinitesimal Approach) . . . . 85
§6. Representations of Semisimple Lie Groups . . . . . . . . . . . . . . . . . . . . . 88
6.1. Semisimple Lie Groups and Algebras . . . . . . . . . . . . . . . . . . . . . 88
6.2. Weights and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3. Representations of Semisimple Lie Groups and Algebras . . . . 91
6.4. Some Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Chapter 6. General Theory of Infinite-Dimensional Unitary
Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
§1. Algebras of Operators in a Hilbert Space and the Decomposition
of Unitary Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
1.1. C* -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
1.2. States and Representations of C* -algebras . . . . . . . . . . . . . . . . 98
1.3. Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
1.4. Direct Integrals of Hilbert Spaces and von Neumann
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
1.5. The Decomposition of Unitary Representations ............ 103
§2. Group Algebras of Locally Compact Groups . . . . . . . . . . . . . . . . . . . 105
2.1. Integration on Groups and Homogeneous Spaces ........... 105
2.2. The Algebras L1(G) and C*(G) ......................... 107
2.3. Unitary Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
§3. Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.1. Topology on the Set of Irreducible Unitary
Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2. Abstract Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.3. Ring Groups and the Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4 A. A. Kirillov
§4. The Theory of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1. Generalized Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2. Infinitesimal Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter 7. The Method of Orbits in the Representation Theory ...... 120
§1. Symplectic Geometry in Homogeneous Spaces ................. 120
1.1. Local Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
1.2. Homogeneous Symplectic Manifolds ...................... 123
1.3. Orbits in the Coadjoint Representation . . . . . . . . . . . . . . . . . . . 126
§2. Representations of Nilpotent Lie Groups . . . . . . . . . . . . . . . . . . . . . . 129
2.1. The Formulation of the Basic Result ..................... 129
2.2. Topology of Gin Terms of Orbits ........................ 134
2.3. The Functors Res and Ind .............................. 136
2.4. Computation of Characters by Orbits . . . . . . . . . . . . . . . . . . . . 137
2.5. Infinitesimal Characters and Orbits . . . . . . . . . . . . . . . . . . . . . . 139
§3. Representations of Solvable Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 140
3.1. Exponential Groups ................................... 140
3.2. General Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
§4. The Method of Orbits for Other Classes of Groups . . . . . . . . . . . . . 146
4.1. Semisimple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.2. General Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.3. Infinite-Dimensional Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.4. Representations of Lie Supergroups and Lie Superalgebras . . 150
Comments on the References .................................... 150
References .................................................... 151
I. Theory of Representations and Noncommutative Harmonic Analysis 5
Chapter 1
A Historical Sketch
§1 . Foreword
Noncommutative harmonic analysis and its basic tool-the theory of group
representations - has existed as an independent domain of mathematics for
about 100 years.
Harmonic analysis in the most general sense can be defined as the math
ematical apparatus applicable to the study and use of symmetry in the sur
rounding world and in its mathematical models. As a rule, the symmetry
is described in terms of transformation groups and the subject of harmonic
analysis is the study of the corresponding representations of groups. Taking
such a point of view, harmonic analysis covers a broad field and includes a
considerable part of the whole of classical and contemporary mathematics. Be
sides this, the idea of a symmetry plays quite a substantial role in theoretical
physics and in other natural sciences.
The full account of the history of harmonic analysis has not yet been written
(see the review papers quoted in the bibliography) and to write it would need
more space, time and qualification than the author has at his disposal. My
task is much more modest - to make the reader familiar with the history of
evolution of the theory of group representations up to the contemporary level.
Understandably, priority is inevitably given to the fields that are the closest
to the author's interests.
§2. Finite-Dimensional Representations
The first ideas of harmonic analysis - the notions of the generating func
tion of a sequence and of "group character" of a finite abelian group - had
appeared in number theory and in probability theory at the beginning of the
eighteenth century, even before the creation of the notion of a group itself.
With the appearance of Fourier series and the Fourier transform in mathemat
ical physics in the nineteenth century, commutative harmonic analysis became
the most powerful tool for the solution of various translation-invariant prob
lems. It is sufficient to recall the theory of differential equations with constant
coefficients, Fourier series and Fourier transform.
The birth of noncommutative harmonic analysis is usually connected with
a series of papers by F.G.Frobenius, published in the period 1896-1901. The
original aim of Frobenius was the solution of a problem posed by his teacher
R.I.W.Dedekind. The problem was the following. Denote by Xt, •.• , Xn ele
ments of a finite group G of order n. The multiplication table (also called the
Cayley table) of the group G can be interpreted as an n x n-matrix; the deter-