Non-Abelian harmonic analysis: applications of SL (2,R)
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1992
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 257 S. graph. Darst. |
| ISBN: | 0387977686 3540977686 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
Notations xi
I Preliminaries
1. Lie Groups and Lie Algebras 1
1.1. Basic Structure 1
1.2. Representations of Lie Groups 9
1.3. Representations of Lie Algebras 20
2. Theory of Fourier Transform 27
2.1. Distributions 27
2.2. Fourier Transform 33
3. Spectral Analysis for Representations of R™ 37
Exercises 41
II Representations of the Lie Algebra of SL(2,R)
1. Standard Modules and the Structure of sl(2) Modules 52
1.1. Indecomposable Modules 52
1.2. Standard Modules 60
1.3. Structure Theorem 64
2. Tensor Products 69
2.1. Tensor Product of Two Lowest Weight Modules 69
2.2. Formal Vectors 72
2.3. Tensor Product Vx ® % 73
3. Formal Eigenvectors 77
3.1. Action of Other Bases of «I(2) 77
3.2. Formal e+ Null Vectors in (Vx ® %)~ 80
3.3. Formal h Eigenvectors in U{y+,v~)~ 81
3.4. Some Modules in U{v+, i/ )~ 84
Exercises 88
xiii
xiv Contents
III Unitary Representations of the Universal Cover of SL(2, R)
1. Infinitesimal Classification 93
1.1. Unitarizability of Standard Modules 93
1.2. A Realization of U(u+, u~) 96
1.3. Unitary Dual of SL(2, E) 99
2. Oscillator Representation 102
2.1. Theory of Hermite Functions 102
2.2. The Contragredient (w *,S(En)*) 107
2.3. Tensor Product wp g w9* 108
2.4. Case q = 0: Theory of Spherical Harmonics 110
Exercises 113
IV Applications to Analysis
1. Bochner s Periodicity Relations ___ 121
1.1. Fourier Transform as an Element of SL(2, R) 121
1.2. Bochner s Periodicity Relations 122
2. Harish Chandra s Restriction Formula 126
2.1. Motivation: Case of 0(3, R) 126
2.2. Harish Chandra s Restriction Formula for U{n) 131
2.3. Some Consequences 138
3. Fundamental Solution of the Laplacian 148
3.1. Fundamental Solution of the Definite Laplacian 149
3.2. Fundamental Solution of the Indefinite Laplacian 152
3.3. Structure of O(p, g) Invariant Distributions Supported on
the Light Cone 162
4. Huygens Principle 164
4.1. The Propagator 165
4.2. Symmetries of the Propagator 167
4.3. Representation Theoretic Considerations 173
4.4. O+(n, 1) Invariant Distributions 175
5. Harish Chandra s Regularity Theorem for SL{2, R), and
the Rossman Harish Chandra Kirillov Character Formula 177
5.1. Regularity of Invariant Eigendistributions 179
5.2. Tempered Distributions and the Character Formula 189
Exercises 195
Contents xv
V Asymptotics of Matrix Coefficients
1. Generalities 204
1.1. Various Decompositions 204
1.2. Matrix Coefficients 206
2. Vanishing of Matrix Coefficients at Infinity for SL(n, R) 211
3. Quantitative Estimates 214
3.1. Decay of Matrix Coefficients of Irreducible Unitary
Representations of SL(2, R) 214
3.2. Decay of Matrix Coefficients of the Regular Representation
ofSL(2,R) 217
3.3. Quantitative Estimates for SL(n,R) 221
4. Some Consequences 229
4.1. Kazhdan s Property T 229
4.2. Ergodic Theory 232
Exercises 236
References 243
Index 253
|
| any_adam_object | 1 |
| author | Howe, Roger 1945- Tan, Eng C. |
| author_GND | (DE-588)171158407 |
| author_facet | Howe, Roger 1945- Tan, Eng C. |
| author_role | aut aut |
| author_sort | Howe, Roger 1945- |
| author_variant | r h rh e c t ec ect |
| building | Verbundindex |
| bvnumber | BV005584186 |
| callnumber-first | Q - Science |
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| dewey-full | 515/.785 |
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| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV005584186 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:31:52Z |
| institution | BVB |
| isbn | 0387977686 3540977686 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003496686 |
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| physical | XV, 257 S. graph. Darst. |
| publishDate | 1992 |
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| spelling | Howe, Roger 1945- Verfasser (DE-588)171158407 aut Non-Abelian harmonic analysis applications of SL (2,R) Roger Howe ; Eng Chye Tan New York u.a. Springer 1992 XV, 257 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Analyse harmonique Analyse harmonique ram Représentations de groupes Représentations de groupes ram Harmonic analysis Representations of groups Darstellung Mathematik (DE-588)4128289-9 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Unimodulare Gruppe (DE-588)4186897-3 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Lineare Gruppe (DE-588)4138778-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 Unimodulare Gruppe (DE-588)4186897-3 s Darstellungstheorie (DE-588)4148816-7 s Darstellung Mathematik (DE-588)4128289-9 s Lineare Gruppe (DE-588)4138778-8 s Tan, Eng C. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003496686&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Howe, Roger 1945- Tan, Eng C. Non-Abelian harmonic analysis applications of SL (2,R) Analyse harmonique Analyse harmonique ram Représentations de groupes Représentations de groupes ram Harmonic analysis Representations of groups Darstellung Mathematik (DE-588)4128289-9 gnd Harmonische Analyse (DE-588)4023453-8 gnd Unimodulare Gruppe (DE-588)4186897-3 gnd Darstellungstheorie (DE-588)4148816-7 gnd Lineare Gruppe (DE-588)4138778-8 gnd |
| subject_GND | (DE-588)4128289-9 (DE-588)4023453-8 (DE-588)4186897-3 (DE-588)4148816-7 (DE-588)4138778-8 |
| title | Non-Abelian harmonic analysis applications of SL (2,R) |
| title_auth | Non-Abelian harmonic analysis applications of SL (2,R) |
| title_exact_search | Non-Abelian harmonic analysis applications of SL (2,R) |
| title_full | Non-Abelian harmonic analysis applications of SL (2,R) Roger Howe ; Eng Chye Tan |
| title_fullStr | Non-Abelian harmonic analysis applications of SL (2,R) Roger Howe ; Eng Chye Tan |
| title_full_unstemmed | Non-Abelian harmonic analysis applications of SL (2,R) Roger Howe ; Eng Chye Tan |
| title_short | Non-Abelian harmonic analysis |
| title_sort | non abelian harmonic analysis applications of sl 2 r |
| title_sub | applications of SL (2,R) |
| topic | Analyse harmonique Analyse harmonique ram Représentations de groupes Représentations de groupes ram Harmonic analysis Representations of groups Darstellung Mathematik (DE-588)4128289-9 gnd Harmonische Analyse (DE-588)4023453-8 gnd Unimodulare Gruppe (DE-588)4186897-3 gnd Darstellungstheorie (DE-588)4148816-7 gnd Lineare Gruppe (DE-588)4138778-8 gnd |
| topic_facet | Analyse harmonique Représentations de groupes Harmonic analysis Representations of groups Darstellung Mathematik Harmonische Analyse Unimodulare Gruppe Darstellungstheorie Lineare Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003496686&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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