Cohomological induction and unitary representations:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, NJ
Princeton Univ. Press
1995
|
| Series: | Princeton mathematical series
45 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 948 S. |
| ISBN: | 0691037566 |
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MARC
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Record in the Search Index
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| adam_text | CONTENTS
Preface xi
Prerequisites by Chapter xv
Standard Notation xvii
INTRODUCTION
1. Origins of Algebraic Representation Theory 3
2. Early Constructions of Representations 7
3. Sections of Homogeneous Vector Bundles 14
4. Zuckerman Functors 23
5. Cohomological Induction 26
6. Hecke Algebra and the Definition of n 30
7. Positivity and the Good Range 32
8. One Dimensional Z and the Fair Range 35
9. Transfer Theorem 36
I. HECKE ALGEBRAS
1. Distributions on Lie Groups 39
2. Hecke Algebras for Compact Lie Groups 45
3. Approximate Identities 60
4. Hecke Algebras in the Group Case 67
5. Abstract Construction 76
6. Hecke Algebras for Pairs (g, K) 87
II. THE CATEGORY C(q, K)
1. Functors P and / 101
2. Properties of P and / 108
3. Constructions within C(g, K) 115
4. Special Properties of P and / in Examples 124
5. Mackey Isomorphisms 145
6. Derived Functors of P and / 156
7. Standard Resolutions 160
8. Koszul Resolution as a Complex 168
9. Reduction of Exactness for the Koszul Resolution 173
10. Exactness in the Abelian Case 176
vii
viii CONTENTS
III. DUALITY THEOREM
1. Easy Duality 181
2. Statement of Hard Duality 182
3. Complexes for Computing P, and I 190
4. Hard Duality as a A Isomorphism 193
5. Proof of g Equivariance in Case (i) 199
6. Motivation for g Equivariance in Case (ii) 215
7. Proof of g Equivariance in Case (ii) 222
8. Proof of Hard Duality in the General Case 227
IV. REDUCTIVE PAIRS
1. Review of Cartan Weyl Theory 231
2. Cartan Weyl Theory for Disconnected Groups 239
3. Reductive Groups and Reductive Pairs 244
4. Cartan Subpairs 248
5. Finite Dimensional Representations 259
6. Parabolic Subpairs 266
7. Harish Chandra Isomorphism 283
8. Infinitesimal Character 297
9. Kostant s Theorem 304
10. Casselman Osborne Theorem 312
11. Algebraic Analog of Bott Borel Weil Theorem 317
V. COHOMOLOGICAL INDUCTION
1. Setting 327
2. Effect on Infinitesimal Character 335
3. Preliminary Lemmas 344
4. Upper Bound on Multiplicities of K Types 347
5. An Euler Poincare Principle for K Types 359
6. Bottom Layer Map 363
7. Vanishing Theorem 369
8. Fundamental Spectral Sequences 379
9. Spectral Sequences for Analysis of K Types 386
10. Hochschild Serre Spectral Sequences 389
11. Composite P Functors and / Functors 399
VI. SIGNATURE THEOREM
1. Setting 401
2. Hermitian Dual and Signature 403
3. Hermitian Duality Relative to P and / 409
4. Statement of Signature Theorem 413
5. Comparison of Shapovalov Forms on K and G 415
6. Preservation of Positivity from L n K to K 421
7. Signature Theorem for £ Badly Disconnected 427
CONTENTS ix
VII. TRANSLATION FUNCTORS
1. Motivation and Examples 435
2. Generalized Infinitesimal Character 441
3. Chevalley s Structure Theorem for Z(g) 449
4. Z(l) Finiteness of u Homology and Cohomology 459
5. Invariants in the Symmetric Algebra 463
6. Kostant s Theory of Harmonics 469
7. Dixmier Duflo Theorem 484
8. Translation Functors 488
9. Integral Dominance 495
10. Overview of Preservation of Irreducibility 509
11. Details of Irreducibility 517
12. Nonvanishing of Certain Translation Functors 524
13. Application to (g, K) Modules with K Connected 528
14. Application to (q, K) Modules with K Disconnected 532
15. Application to Cohomological Induction 542
16. Application to u Homology and Cohomology 546
VIII. IRREDUCIBILITY THEOREM
1. Main Theorem and Overview 549
2. Proof of Irreducibility Theorem 553
3. Role of Integral Dominance 559
4. Irreducibility Theorem for AT Badly Disconnected 561
5. Consideration of Aq (A.) 569
IX. UNITARIZABILITY THEOREM
1. Statement of Theorem 597
2. Signature Character and Examples 600
3. Signature Character of ind Z* 611
4. Signature Character of Alternating Tensors 623
5. Signature Character and Formal Character of CS{Z) 624
6. Improved Theorem for Aq (X) 630
X. MINIMAL K TYPES
1. Admissibility of Irreducible (g, K) Modules 634
2. Minimal K Types and Infinitesimal Characters 641
3. Minimal K Types and Cohomological Induction 650
XI. TRANSFER THEOREM 652
1. Parabolic Induction Globally 653
2. Parabolic Induction Infinitesimally 667
3. Preliminary Lemmas 678
4. Spectral Sequences for Induction in Stages 680
5. Transfer Theorem 684
X CONTENTS
6. Standard Modules 697
7. Normalization of C and 11 713
8. Discrete Series and Limits 730
9. Langlands Parameters 739
10. Cohomological Induction and Standard Modules 757
11. Minimal K Type Formula 766
12. Cohomological Induction and Minimal # Types 781
XII. EPILOG:
WEAKLY UNIPOTENT REPRESENTATIONS 791
APPENDICES
A. Miscellaneous Algebra
1. Good Categories 803
2. Completely Reducible Modules 806
3. Modules of Finite Length 814
4. Grothendieck Group 818
B. Distributions on Manifolds
1. Topology on C°°(X) 820
2. Distributions and Support 824
3. Fubini s Theorem 828
4. Distributions Supported on Submanifolds 832
C. Elementary Homological Algebra
1. Projectives and Injectives 836
2. Functors 839
3. Derived Functors 843
4. Long Exact Sequences 845
5. Euler Poincare Principle 849
D. Spectral Sequences
1. Spectral Sequence of a Filtered Complex 855
2. Spectral Sequences of a Double Complex 872
3. Derived Functors of a Composition 879
4. Derived Functors of a Filtered Module 887
Notes 891
References 919
Index of Notation 933
Index 941
|
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| indexdate | 2024-07-09T17:49:51Z |
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| isbn | 0691037566 |
| language | English |
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| physical | XVII, 948 S. |
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| series2 | Princeton mathematical series |
| spelling | Knapp, Anthony W. 1941- Verfasser (DE-588)132959690 aut Cohomological induction and unitary representations Anthony W. Knapp and David A. Vogan Princeton, NJ Princeton Univ. Press 1995 XVII, 948 S. txt rdacontent n rdamedia nc rdacarrier Princeton mathematical series 45 Halbeinfache Lie-Gruppe (DE-588)4122188-6 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Homologietheorie (DE-588)4141714-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Halbeinfache Lie-Gruppe (DE-588)4122188-6 s Darstellungstheorie (DE-588)4148816-7 s Homologietheorie (DE-588)4141714-8 s Harmonische Analyse (DE-588)4023453-8 s DE-604 Vogan, David A. Verfasser aut Princeton mathematical series 45 (DE-604)BV000019035 45 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006842909&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Knapp, Anthony W. 1941- Vogan, David A. Cohomological induction and unitary representations Princeton mathematical series Halbeinfache Lie-Gruppe (DE-588)4122188-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Homologietheorie (DE-588)4141714-8 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4122188-6 (DE-588)4148816-7 (DE-588)4141714-8 (DE-588)4023453-8 |
| title | Cohomological induction and unitary representations |
| title_auth | Cohomological induction and unitary representations |
| title_exact_search | Cohomological induction and unitary representations |
| title_full | Cohomological induction and unitary representations Anthony W. Knapp and David A. Vogan |
| title_fullStr | Cohomological induction and unitary representations Anthony W. Knapp and David A. Vogan |
| title_full_unstemmed | Cohomological induction and unitary representations Anthony W. Knapp and David A. Vogan |
| title_short | Cohomological induction and unitary representations |
| title_sort | cohomological induction and unitary representations |
| topic | Halbeinfache Lie-Gruppe (DE-588)4122188-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Homologietheorie (DE-588)4141714-8 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Halbeinfache Lie-Gruppe Darstellungstheorie Homologietheorie Harmonische Analyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006842909&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000019035 |
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