Representation Theory of Semisimple Groups: An Overview Based on Examples (PMS-36)

by Anthony W. Knapp

Overview

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Product Details

ISBN-13:	9780691090894
Publisher:	Princeton University Press
Publication date:	10/07/2001
Series:	Princeton Mathematical Series , #89
Edition description:	With a New preface by the author
Pages:	800
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Anthony W. Knapp is Emeritus Professor of Mathematics, State University of New York at Stony Brook. The author of numerous books, he is the former editor of the Notices of the American Mathematical Society.

Table of Contents

	Preface to the Princeton Landmarks in Mathematics Edition	xiii
	Preface	xv
	Acknowledgments	xix
Chapter I.	Scope of the Theory
1.	The Classical Groups	3
2.	Cartan Decomposition	7
3.	Representations	10
4.	Concrete Problems in Representation Theory	14
5.	Abstract Theory for Compact Groups	14
6.	Application of the Abstract Theory to Lie Groups	23
7.	Problems	24
Chapter II.	Representations of SU(2), SL(2, R), and SL(2, C)
1.	The Unitary Trick	28
2.	Irreducible Finite-Dimensional Complex-Linear Representations of sl(2, C)	30
3.	Finite-Dimensional Representations of sl(2, C)	31
4.	Irreducible Unitary Representations of SL(2, C)	33
5.	Irreducible Unitary Representations of SL(2, R)	35
6.	Use of SU(1, 1)	39
7.	Plancherel Formula	41
8.	Problems	42
Chapter III.	C[superscript [infinity] Vectors and the Universal Enveloping Algebra
1.	Universal Enveloping Algebra	46
2.	Actions on Universal Enveloping Algebra	50
3.	C[superscript [infinity] Vectors	51
4.	Garding Subspace	55
5.	Problems	57
Chapter IV.	Representations of Compact Lie Groups
1.	Examples of Root Space Decompositions	60
2.	Roots	65
3.	Abstract Root Systems and Positivity	72
4.	Weyl Group, Algebraically	78
5.	Weights and Integral Forms	81
6.	Centalizers of Tori	86
7.	Theorem of the Highest Weight	89
8.	Verma Modules	93
9.	Weyl Group, Analytically	100
10.	Weyl Character Formula	104
11.	Problems	109
Chapter V.	Structure Theory for Noncompact Groups
1.	Cartan Decomposition and the Unitary Trick	113
2.	Iwasawa Decomposition	116
3.	Regular Elements, Weyl Chambers, and the Weyl Group	121
4.	Other Decompositions	126
5.	Parabolic Subgroups	132
6.	Integral Formulas	137
7.	Borel-Weil Theorem	142
8.	Problems	147
Chapter VI.	Holomorphic Discrete Series
1.	Holomorphic Discrete Series for SU(1, 1)	150
2.	Classical Bounded Symmetric Domains	152
3.	Harish-Chandra Decomposition	153
4.	Holomorphic Discrete Series	158
5.	Finiteness of an Integral	161
6.	Problems	164
Chapter VII.	Induced Representations
1.	Three Pictures	167
2.	Elementary Properties	169
3.	Bruhat Theory	172
4.	Formal Intertwining Operators	174
5.	Gindikin-Karpelevic Formula	177
6.	Estimates on Intertwining Operators, Part I	181
7.	Analytic Continuation of Intertwining Operators, Part I	183
8.	Spherical Functions	185
9.	Finite-Dimensional Representations and the H function	191
10.	Estimates on Intertwining Operators, Part II	196
11.	Tempered Representations and Langlands Quotients	198
12.	Problems	201
Chapter VIII.	Admissible Representations
1.	Motivation	203
2.	Admissible Representations	205
3.	Invariant Subspaces	209
4.	Framework for Studying Matrix Coefficients	215
5.	Harish-Chandra Homomorphism	218
6.	Infinitesimal Character	223
7.	Differential Equations Satisfied by Matrix Coefficients	226
8.	Asymptotic Expansions and Leading Exponents	234
9.	First Application: Subrepresentation Theorem	238
10.	Second Application: Analytic Continuation of Interwining Operators, Part II	239
11.	Third Application: Control of K-Finite Z(g[superscript C])-Finite Functions	242
12.	Asymptotic Expansions near the Walls	247
13.	Fourth Application: Asymptotic Size of Matrix Coefficients	253
14.	Fifth Application: Identification of Irreducible Tempered Representations	258
15.	Sixth Application: Langlands Classification of Irreducible Admissible Representations	266
16.	Problems	276
Chapter IX.	Construction of Discrete Series
1.	Infinitesimally Unitary Representations	281
2.	A Third Way of Treating Admissible Representations	282
3.	Equivalent Definitions of Discrete Series	284
4.	Motivation in General and the Construction in SU(1, 1)	287
5.	Finite-Dimensional Spherical Representations	300
6.	Duality in the General Case	303
7.	Construction of Discrete Series	309
8.	Limitations on K Types	320
9.	Lemma on Linear Independence	328
10.	Problems	330
Chapter X.	Global Characters
1.	Existence	333
2.	Character Formulas for SL(2, R)	338
3.	Induced Characters	347
4.	Differential Equations	354
5.	Analyticity on the Regular Set, Overview and Example	355
6.	Analyticity on the Regular Set, General Case	360
7.	Formula on the Regular Set	368
8.	Behavior on the Singular Set	371
9.	Families of Admissible Representations	374
10.	Problems	383
Chapter XI.	Introduction to Plancherel Formula
1.	Constructive Proof for SU(2)	385
2.	Constructive Proof for SL(2, C)	387
3.	Constructive Proof for SL(2, R)	394
4.	Ingredients of Proof for General Case	401
5.	Scheme of Proof for General Case	404
6.	Properties of F[subscript f]	407
7.	Hirai's Patching Conditions	421
8.	Problems	425
Chapter XII.	Exhaustion of Discrete Series
1.	Boundedness of Numerators of Characters	426
2.	Use of Patching Conditions	432
3.	Formula for Discrete Series Characters	436
4.	Schwartz Space	447
5.	Exhaustion of Discrete Series	452
6.	Tempered Distributions	456
7.	Limits of Discrete Series	460
8.	Discrete Series of M	467
9.	Schmid's Identity	473
10.	Problems	476
Chapter XIII.	Plancherel Formula
1.	Ideas and Ingredients	482
2.	Real-Rank-One Groups	482
3.	Real-Rank-One Groups, Part II	485
4.	Averaged Discrete Series	494
5.	Sp (2, R)	502
6.	General Case	511
7.	Problems	512
Chapter XIV.	Irreducible Tempered Representations
1.	SL(2, R) from a More General Point of View	515
2.	Eisenstein Integrals	520
3.	Asymptotics of Eisenstein Integrals	526
4.	The [eta] Functions for Intertwining Operators	535
5.	First Irreducibility Results	540
6.	Normalization of Intertwining Operators and Reducibility	543
7.	Connection with Plancherel Formula when dim A = 1	547
8.	Harish-Chandra's Completeness Theorem	553
9.	R Group	560
10.	Action by Weyl Group on Representations of M	568
11.	Multiplicity One Theorem	577
12.	Zuckerman Tensoring of Induced Representations	584
13.	Generalized Schmid Identities	587
14.	Inversion of Generalized Schmid Identities	595
15.	Complete Reduction of Induced Representations	599
16.	Classification	606
17.	Revised Langlands Classification	614
18.	Problems	621
Chapter XV.	Minimal K Types
1.	Definition and Formula	626
2.	Inversion Problem	635
3.	Connection with Intertwining Operators	641
4.	Problems	647
Chapter XVI.	Unitary Representations
1.	SL(2, R) and SL(2, C)	650
2.	Continuity Arguments and Complementary Series	653
3.	Criterion for Unitary Representations	655
4.	Reduction to Real Infinitesimal Character	660
5.	Problems	665
Appendix A	Elementary Theory of Lie Groups
1.	Lie Algebras	667
2.	Structure Theory of Lie Algebras	668
3.	Fundamental Group and Covering Spaces	670
4.	Topological Groups	673
5.	Vector Fields and Submanifolds	674
6.	Lie Groups	679
Appendix B	Regular Singular Points of Partial Differential Equations
1.	Summary of Classical One-Variable Theory	685
2.	Uniqueness and Analytic Continuation of Solutions in Several Variables	690
3.	Analog of Fundamental Matrix	693
4.	Regular Singularities	697
5.	Systems of Higher Order	700
6.	Leading Exponents and the Analog of the Indicial Equation	703
7.	Uniqueness of Representation	710
Appendix C	Roots and Restricted Roots for Classical Groups
1.	Complex Groups	713
2.	Noncompact Real Groups	713
3.	Roots vs. Restricted Roots in Noncompact Real Groups	715
	Notes	719
	References	747
	Index of Notation	763
	Index	767

Editorial Reviews

Winner of the 1997 Leroy P. Steele Prize, American Mathematics Society

From the Publisher

Each [theme] is developed carefully and thoroughly, with beautifully worked examples and proofs that reflect long experience in teaching and research. . . . This result is delightful: a readable text that loses almost none of its value as a reference work.

Bulletin of the American Mathematical Society - David A. Vogan Jr.

Anthony Knapp has written a marvelous book. . . . Written with accuracy, style, and a genuine desire to communicate the materials. . . . This is one of the finest books I have ever had the pleasure to read, and I recommend it in the strongest possible terms to anyone wishing to appreciate the intricate beauty and technical difficulty of representation theory of semisimple Lie groups.

Bulletin of the London Mathematical Society - R. J. Plymen

Anthony Knapp has written a marvelous book. . . . Written with accuracy, style, and a genuine desire to communicate the materials. . . . This is one of the finest books I have ever had the pleasure to read, and I recommend it in the strongest possible terms to anyone wishing to appreciate the intricate beauty and technical difficulty of representation theory of semisimple Lie groups.

Bulletin of the London Mathematical Society - R.J. Plymen

Each [theme] is developed carefully and thoroughly, with beautifully worked examples and proofs that reflect long experience in teaching and research. . . . This result is delightful: a readable text that loses almost none of its value as a reference work.
— David A. Vogan Jr.

Bulletin of the American Mathematical Society

Anthony Knapp has written a marvelous book. . . . Written with accuracy, style, and a genuine desire to communicate the materials. . . . This is one of the finest books I have ever had the pleasure to read, and I recommend it in the strongest possible terms to anyone wishing to appreciate the intricate beauty and technical difficulty of representation theory of semisimple Lie groups.
— R. J. Plymen

Bulletin of the London Mathematical Society