Spectral decomposition and Eisenstein series: une paraphrase de l'écriture
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge ; New York ; Melbourne ; Madrid ; Cape Town ; Singapor ; São Paulo
Cambridge University Press
1995
|
| Series: | Cambridge tracts in mathematics
113 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | xxvii, 338 Seiten |
| ISBN: | 9780521070355 9780521418935 0521418933 |
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MARC
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| 240 | 1 | 0 | |a Décomposition spectrale et séries d'Eisenstein |
| 245 | 1 | 0 | |a Spectral decomposition and Eisenstein series |b une paraphrase de l'écriture |c C. Moeglin, J.-L. Waldspurger (Université de Paris VII) |
| 264 | 1 | |a Cambridge ; New York ; Melbourne ; Madrid ; Cape Town ; Singapor ; São Paulo |b Cambridge University Press |c 1995 | |
| 264 | 4 | |c © 1995 | |
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Cambridge tracts in mathematics |v 113 | |
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| 650 | 7 | |a Décomposition (Mathématiques) |2 ram | |
| 650 | 7 | |a Eisenstein, séries d' |2 ram | |
| 650 | 7 | |a Eisenstein-reeksen |2 gtt | |
| 650 | 7 | |a Formes automorphiques |2 ram | |
| 650 | 7 | |a Spectraaltheorie |2 gtt | |
| 650 | 7 | |a Théorie spectrale (Mathématiques) |2 ram | |
| 650 | 4 | |a Automorphic forms | |
| 650 | 4 | |a Decomposition (Mathematics) | |
| 650 | 4 | |a Eisenstein series | |
| 650 | 4 | |a Spectral theory (Mathematics) | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preamble xiii
Notation xxv
I Hypotheses, automorphic forms, constant terms
1.1 Hypotheses and general notation 1
1.1.1 Definitions 1
1.1.2 Description of G 2
1.1.3 The covering GV— G(A); hypotheses and properties 3
1.1.4 Levi subgroups and characters 4
1.1.5 Description of G continued 8
1.1.6 Roots and coroots 10
1.1.7 The Weyl group 12
1.1.8 Decomposition into elementary symmetries 14
1.1.9 A composition lemma 15
1.1.10 Decomposition of Re aM 15
1.1.11 Return to the coroots 16
1.1.12 The scalar product 17
1.1.13 Measures 18
1.2 Automorphic forms: growth, constant terms 18
1.2.1 Siegel domains 18
1.2.2 Heights 20
1.2.3 Functions with moderate growth 24
vi Contents
1.2.4 An upper bound lemma 25
1.2.5 Construction of functions with moderate growth 26
1.2.6 Constant terms 27
1.2.7 Approximation of a function by its constant term,
function field case 27
1.2.8 Corollary 28
1.2.9 Corollary 29
1.2.10 The number field case 30
1.2.11 Corollary 34
1.2.12 Rapidly decreasing functions 34
1.2.13 Truncation 35
1.2.14 Lemma 36
1.2.15 Lemma 36
1.2.16 Lemma 36
1.2.17 Automorphic forms 37
1.2.18 Cuspidal forms 39
1.3 Cuspidal components 40
1.3.1 ^M finite functions 40
1.3.2 Decomposition of an automorphic form 41
1.3.3 Exponents of cuspidal automorphic forms 43
1.3.4 Cuspidal components 45
1.3.5 Cuspidal components and automorphic functions 47
1.3.6 The function field case: independence of the choice of vq 48
1.4 Upper bounds as functions of the constant term 49
1.4.1 Lemma 49
1.4.2 Exponential polynomial functions 51
1.4.3 Uniform upper bounds for automorphic forms 56
1.4.4 Sequences of automorphic forms 59
1.4.5 Proof of I.4.4(a) 60
1.4.6 Partial proof of I.4.4(c) 62
1.4.7 Proof of I.4.4(b) 63
1.4.8 End of the proof of 1.4.4 66
1.4.9 Frechet spaces and holomorphic functions 66
1.4.10 Automorphic forms depending holomorphically on a
parameter 68
1.4.11 Exponents of square integrable automorphic forms 74
Contents vii
II Decomposition according to cuspidal data
III Definitions 78
II. 1.1 Equivalence classes of irreducible subrepresentations
of the space of automorphic forms and complex
analytic structure 78
II. 1.2 Paley Wiener functions 80
II.1.3 Fourier transforms 81
II. 1.4 The spaces P,^,, 84
II. 1.5 Eisenstein series 85
II. 1.6 Intertwining operators 87
11.1.7 Constant terms of Eisenstein series 91
11.1.8 Adjunction of intertwining operators 93
II. 1.9 Euler expansions of intertwining operators 94
II.1.10 Pseudo Eisenstein series 95
II. 1.11 Relation between pseudo Eisenstein series and
Eisenstein series 99
II.1.12 Density of pseudo Eisenstein series 99
11.2 Calculation of the scalar product of two pseudo
Eisenstein series 100
11.2.1 Statement 100
11.2.2 Calculation of constant terms of pseudo
Eisenstein series 101
11.2.3 End of the calculation of the scalar product 105
11.2.4 Decomposition of L2(G(/c) G){ along
the cuspidal support 106
11.2.5 Lemma 107
III Hilbertian operators and automorphic forms
III.l Hilbertian operators 109
111.1.1 A family of operators 109
111.1.2 Operations on pseudo Eisenstein series 110
111.1.3 Transformations of Hilbert spaces 111
III. 1.4 Bounded operators 112
III.1.5 Adjunction of non bounded operators 112
III. 1.6 Spectral projections 113
III.1.7 Transformations of L2(G(k) G^ 115
viii Contents
111.2 A decomposition of the space of automorphic forms 115
111.2.1 Operators on the space of automorphic forms 115
111.2.2 Cuspidal components 116
111.2.3 Remark 120
111.2.4 A subalgebra of operators 120
111.2.5 Proof of Proposition III.2.1 123
111.2.6 Decomposition of the space of automorphic forms 128
111.3 Cuspidal exponents and square integrable
automorphic forms 129
111.3.1 The result 129
111.3.2 Definition of N(G) 130
111.3.3 Reduction of the problem 131
111.3.4 Proof of Proposition III.3.1 131
IV Continuation of Eisenstein series
IV. 1 The results 135
IV.1.1 The spaces 135
IV.1.2 The representations 136
IV.1.3 Holomorphic functions 136
IV.1.4 Holomorphic operators 137
IV.1.5 Rationality 138
IV.1.6 Singularities along hyperplanes 138
IV.1.7 Holomorphic property of Eisenstein series 138
IV.1.8 Continuation of Eisenstein series 140
IV.1.9 Properties of Eisenstein series 140
IV.1.10 The functional equation 141
IV.1.11 Singularities of Eisenstein series 141
IV.1.12 The function field case 141
IV.2 Some preparations 141
IV.2.1 Transformation of the problem 141
IV.2.2 Choice of functions in the Hecke algebra 142
IV.2.3 Compact operators 143
IV.2.4 Truncation is an orthogonal projection 144
IV.2.5 Upper bound of a truncated kernel 145
IV.2.6 Compactness of truncated operators 147
IV.2.7 A geometric lemma 147
IV.2.8 Remark 148
Contents ix
IV.3 The case of relative rank 1 149
IV.3.1 The situation of relative rank 1 149
IV.3.2 An auxiliary series 149
IV.3.3 Truncation of auxiliary series 153
IV.3.4 The functional equation for truncation of
Eisenstein series 153
IV.3.5 Resolution of a functional equation 154
IV.3.6 A decomposition of Eisenstein series 156
IV.3.7 Corollary 156
IV.3.8 An injectivity lemma 156
IV.3.9 Proof of Theorem IV. 1.8 157
IV.3.10 Proof of Proposition IV. 1.9 158
IV.3.11 Proof of Theorem I V.I. 10 160
IV.3.12 Proof of Proposition I V.I. 11 161
IV.3.13 Proof of Proposition IV.1.12(b) 162
IV.3.14 Proof of Proposition IV.1.12(a) 163
IV.4 The general case 165
IV.4.1 Continuation of intertwining operators 165
IV.4.2 The auxiliary series 165
IV.4.3 Continuation of Eisenstein series 166
IV.4.4 End of the proof 167
IV.4.5 Remark 168
V Construction of the discrete spectrum via
residues
V.I Generalities and the residue theorem 169
V.I.I Affine subspace 169
V.1.2 Meromorphic functions with polynomial singularities
on 3h 171
V.I.3 Residue data 173
V.1.4 Remark on the choices made in V.I.3 174
V.1.5 The residue theorem 175
V.2 Decomposition of the scalar product of two pseudo
Eisenstein series 180
V.2.1 Notation 180
V.2.2 Statement of the theorem 182
V.2.3 Beginning of the proof: first step 183
V.2.4 Second step 185
V.2.5 Corollary 186
x Contents
V.2.6 Corollary 187
V.2.7 End of the proof of V.2.2 187
V.2.8 Remark on the number field case 188
V.2.9 Lemma 188
V.2.10 Lemma 191
V.2.11 Sublemma 192
V.3 Decomposition along the spectrum of the
operators A(/) 195
V.3.1 Hypotheses and notation for this part 196
V.3.2 Statement 197
V.3.3 Definition of the family of projections 199
V.3.4 Plan and beginning of the proof 200
V.3.5 Lemma 202
V.3.6(a) Remark 203
V.3.6(b) Remark 207
V.3.7 Proof of the assertions of V.3.2(4) for every j e Pgf, 208
V.3.8 Another lemma 209
V.3.9 Proof of V.3.2(5)(i) for general n 210
V.3.10 Remark 214
V.3.11 Spectral decomposition of L^T 214
V.3.12 Proof of V.3.2 217
V.3.13 Decomposition of qTL2(G(k) G)x^ 221
V.3.14 Decomposition of L2(G(/c) G)x 226
V.3.15 Adjunction formula for residues of intertwining
operators 227
V.3.16 Cuspidal exponents of residues of Eisenstein series 229
V.3.17 Generalisation: decomposition of L2(UL(A)L(k) G)iL 230
VI Spectral decomposition via the discrete Levi spectrum
VI. 1 Discrete parameter 234
VI. 1.1 The standard Levi associated with an element
of Sx 234
VI. 1.2 Singularities of residues of Eisenstein series 234
VI. 1.3 Corollary 240
VI. 1.4 Reduction to the Levi subgroup 242
VI. 1.5 Functional equation for residues of Eisenstein series 250
VI. 1.6 Positivity property of the origin of singular planes 251
VI. 1.7 Association classes of standard Levis attached to an
element of [Sx] 255
Contents xi
VI. 1.8 Reality of the origin of elements of SingG 256
VI. 1.9 Discrete parameter 257
VI.2 Spectral decomposition 259
VI.2.1 The most general statement 259
VI.2.2 A more technical but more precise statement 260
VI.2.3 Proof of VI.2.l(i) 263
VI.2.4 Description of A^n (for general n) 265
VI.2.5 Proof of the spectral decomposition of VI.2.2 (number
field case) 269
VI.2.6 Proof of the spectral decomposition of VI.2.1 (number
field case) 271
VI.2.7 The function field case 272
Appendix I Lifting of unipotent subgroups into a
central extension 273
Appendix II Automorphic forms and Eisenstein series
over a function field 278
1. Derivatives of Eisenstein series 278
1.1 Some reminders 278
1.2 Eisenstein series 279
1.3 Derivations 279
1.4 Derivatives of Eisenstein series 280
1.5 Function fields 280
2. Relations between constant terms of
automorphic forms 281
2.1 Notation 281
2.2 Calculation of a scalar product 282
2.3 Vanishing of a pseudo Eisenstein series 284
2.4 The global sections generate the local sections 285
2.5 A linear system 286
2.6 Resolution of the system 287
2.7 Expression of an automorphic form as the derivative of
an Eisenstein series 290
2.8 Theorem 292
2.9 Remarks 292
xii Contents
3. Proof of Proposition 2.4 293
3.1 Transformation of the problem 293
3.2 Calculation in the completion 294
3.3 End of the proof 295
Appendix III On the discrete spectrum of G2 298
1. The result 298
2. Explicit calculations for G2 299
3. Reminder of the method 302
4. Calculation of a residue 303
5. Calculation continued 305
6. Expression of the scalar product 307
7. The Hecke algebra 308
8. Intertwining operators 310
9. The final calculation 312
Appendix IV Non connected groups 314
1. Definitions 314
2. Levi subgroups 315
3. The Weyl group 317
4. The compact subgroup 318
5. Coverings, centres 319
6. Automorphic forms 320
7. Cuspidal components 322
8. Eisenstein series 323
9. The functions trip and trivialisation of fibre bundles 324
10. Paley Wiener functions, P(m,^) and P(^ p( 327
11. Pseudo Eisenstein series 327
12. Constant terms of pseudo Eisenstein series;
scalar product 328
13. The operators A(/) 330
14. Spectral decomposition 331
Bibliography 334
Index 337
|
| any_adam_object | 1 |
| author | Moeglin, Colette 1953- Waldspurger, Jean-Loup 1953- |
| author_GND | (DE-588)111560519 (DE-588)135928125 |
| author_facet | Moeglin, Colette 1953- Waldspurger, Jean-Loup 1953- |
| author_role | aut aut |
| author_sort | Moeglin, Colette 1953- |
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| building | Verbundindex |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 SK 620 |
| ctrlnum | (OCoLC)30353905 (DE-599)BVBBV010587194 |
| dewey-full | 515/.243 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.243 |
| dewey-search | 515/.243 |
| dewey-sort | 3515 3243 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010587194 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:55:30Z |
| institution | BVB |
| isbn | 9780521070355 9780521418935 0521418933 |
| language | English |
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| physical | xxvii, 338 Seiten |
| publishDate | 1995 |
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| publisher | Cambridge University Press |
| record_format | marc |
| series | Cambridge tracts in mathematics |
| series2 | Cambridge tracts in mathematics |
| spelling | Moeglin, Colette 1953- Verfasser (DE-588)111560519 aut Décomposition spectrale et séries d'Eisenstein Spectral decomposition and Eisenstein series une paraphrase de l'écriture C. Moeglin, J.-L. Waldspurger (Université de Paris VII) Cambridge ; New York ; Melbourne ; Madrid ; Cape Town ; Singapor ; São Paulo Cambridge University Press 1995 © 1995 xxvii, 338 Seiten txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 113 Hier auch später erschienene, unveränderte Nachdrucke Automorfe functies gtt Décomposition (Mathématiques) ram Eisenstein, séries d' ram Eisenstein-reeksen gtt Formes automorphiques ram Spectraaltheorie gtt Théorie spectrale (Mathématiques) ram Automorphic forms Decomposition (Mathematics) Eisenstein series Spectral theory (Mathematics) Eisenstein-Reihe (DE-588)4131762-2 gnd rswk-swf Spektraldarstellung (DE-588)4182162-2 gnd rswk-swf Spektraldarstellung (DE-588)4182162-2 s Eisenstein-Reihe (DE-588)4131762-2 s DE-604 Waldspurger, Jean-Loup 1953- Verfasser (DE-588)135928125 aut Erscheint auch als Online-Ausgabe 978-0-511-47090-5 Cambridge tracts in mathematics 113 (DE-604)BV000000001 113 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007059389&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moeglin, Colette 1953- Waldspurger, Jean-Loup 1953- Spectral decomposition and Eisenstein series une paraphrase de l'écriture Cambridge tracts in mathematics Automorfe functies gtt Décomposition (Mathématiques) ram Eisenstein, séries d' ram Eisenstein-reeksen gtt Formes automorphiques ram Spectraaltheorie gtt Théorie spectrale (Mathématiques) ram Automorphic forms Decomposition (Mathematics) Eisenstein series Spectral theory (Mathematics) Eisenstein-Reihe (DE-588)4131762-2 gnd Spektraldarstellung (DE-588)4182162-2 gnd |
| subject_GND | (DE-588)4131762-2 (DE-588)4182162-2 |
| title | Spectral decomposition and Eisenstein series une paraphrase de l'écriture |
| title_alt | Décomposition spectrale et séries d'Eisenstein |
| title_auth | Spectral decomposition and Eisenstein series une paraphrase de l'écriture |
| title_exact_search | Spectral decomposition and Eisenstein series une paraphrase de l'écriture |
| title_full | Spectral decomposition and Eisenstein series une paraphrase de l'écriture C. Moeglin, J.-L. Waldspurger (Université de Paris VII) |
| title_fullStr | Spectral decomposition and Eisenstein series une paraphrase de l'écriture C. Moeglin, J.-L. Waldspurger (Université de Paris VII) |
| title_full_unstemmed | Spectral decomposition and Eisenstein series une paraphrase de l'écriture C. Moeglin, J.-L. Waldspurger (Université de Paris VII) |
| title_short | Spectral decomposition and Eisenstein series |
| title_sort | spectral decomposition and eisenstein series une paraphrase de l ecriture |
| title_sub | une paraphrase de l'écriture |
| topic | Automorfe functies gtt Décomposition (Mathématiques) ram Eisenstein, séries d' ram Eisenstein-reeksen gtt Formes automorphiques ram Spectraaltheorie gtt Théorie spectrale (Mathématiques) ram Automorphic forms Decomposition (Mathematics) Eisenstein series Spectral theory (Mathematics) Eisenstein-Reihe (DE-588)4131762-2 gnd Spektraldarstellung (DE-588)4182162-2 gnd |
| topic_facet | Automorfe functies Décomposition (Mathématiques) Eisenstein, séries d' Eisenstein-reeksen Formes automorphiques Spectraaltheorie Théorie spectrale (Mathématiques) Automorphic forms Decomposition (Mathematics) Eisenstein series Spectral theory (Mathematics) Eisenstein-Reihe Spektraldarstellung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007059389&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000001 |
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