Representation Theory of Semisimple Groups: An Overview Based on Examples

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Princeton University Press, 2001 - Mathematics - 773 pages
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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.

  

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Contents

REPRESENTATIONS OF SU2 SL2 R
28
C VECTORS AND THE UNIVERSAL
46
4 Girding Subspace
55
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
8 Behavior on the Singular Set
371
9 Families of Admissible Representations
374
10 Problems
383
INTRODUCTION TO PLANCHEREL FORMULA 1 Constructive Proof for SU2
385
2 Constructive Proof for SL2 C
387
3 Constructive Proof for SL2 R
394
4 Ingredients of Proof for General Case
401
5 Scheme of Proof for General Case
404

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
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 BorelWeil Theorem
142
8 Problems
147
HOLOMORPHIC DISCRETE SERIES 1 Holomorphic Discrete Series for SU11
150
2 Classical Bounded Symmetric Domains
152
3 HarishChandra Decomposition
153
4 Holomorphic Discrete Series
158
5 Finiteness of an Integral
161
6 Problems
164
INDUCED REPRESENTATIONS 1 Three Pictures
167
2 Elementary Properties
169
3 Bruhat Theory
172
4 Formal Intertwining Operators
174
5 iindikinKarpelevic Formula
177
6 Estimates on Intertwining Operators Part I
181
7 Analytic Continuation of Intertwining Operators Part I
183
8 Spherical Functions
185
9 FiniteDimensional Representations and the H function
191
10 Estimates on Intertwining Operators Part II
196
11 Tempered Representations and Langlands Quotients
198
12 Problems
201
ADMISSIBLE REPRESENTATIONS 1 Motivation
203
2 Admissible Representations
205
3 Invariant Subspaces
209
4 Framework for Studying Matrix Coefficients
215
5 HarishChandra Homomorphism
218
6 Infinitesimal Character
223
7 Differential Equations Satisfied by Matrix Coefficients
226
8 Asymptotic Expansions and Leading Exponents
234
Subrepresentation Theorem
238
Analytic Continuation of Interwining Operators Part II
239
Control of KFinite ZgcFinite Functions
242
12 Asymptotic Expansions near the Walls
247
Asymptotic Size of Matrix Coefficients
253
Identification of Irreducible Tempered Representations
258
Langlands Classification of Irreducible Admissible Representations
266
16 Problems
276
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 SU11
287
5 FiniteDimensional 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
GLOBAL CHARACTERS 1 Existence
333
2 Character Formulas for SL2 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
6 Properties of Ff
407
7 Hirais Patching Conditions
421
8 Problems
425
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 Schmids Identity
473
10 Problems
476
PLANCHEREL FORMULA 1 Ideas and Ingredients
482
3 RealRankOne Groups Part II
485
4 Averaged Discrete Series
494
5 Sp2R
502
6 General Case
511
7 Problems
512
IRREDUCIBLE TEMPERED REPRESENTATIONS 1 SL2 R from a More General Point of View
515
2 Eisenstein Integrals
520
3 Asymptotics of Eisenstein Integrals
526
4 The r 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 HarishChandras 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
MINIMAL K TYPES 1 Definition and Formula
626
2 Inversion Problem
635
3 Connection with Intertwining Operators
641
4 Problems
647
UNITARY REPRESENTATIONS 1 SL2RandSL2C
650
2 Continuity Arguments and Complementary Series
653
3 Criterion for Unitary Representations
655
4 Reduction to Real Infinitesimal Character
660
5 Problems
665
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
REGULAR SINGULAR POINTS OF PARTIAL DIFFERENTIAL EQUATIONS 1 Summary of Classical OneVariable 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
ROOTS AND RESTRICTED ROOTS FOR CLASSICAL
713
NOTES
719
REFERENCES
747
INDEX OF NOTATION
763
Copyright

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Elliptic Curves
Anthony W. Knapp
Limited preview - 1992
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