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