Double affine Hecke algebras
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| Format: | Buch |
| Sprache: | English |
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Cambridge University Press
2005
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| Schriftenreihe: | London Mathematical Society lecture note series
319 |
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| 100 | 1 | |a Tscherednik, Iwan Wladimirowitsch |d 1951- |e Verfasser |0 (DE-588)1104405687 |4 aut | |
| 245 | 1 | 0 | |a Double affine Hecke algebras |c Ivan Cherednik |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2005 | |
| 300 | |a XII, 434 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a London Mathematical Society lecture note series |v 319 | |
| 650 | 7 | |a Affiene algebra's |2 gtt | |
| 650 | 4 | |a Analyse harmonique | |
| 650 | 4 | |a Groupes algébriques affines | |
| 650 | 7 | |a Harmonische analyse |2 gtt | |
| 650 | 4 | |a Hecke, Algèbres de | |
| 650 | 4 | |a Knizhnik-Zamolodchikov, Équations de | |
| 650 | 4 | |a Polynômes orthogonaux | |
| 650 | 4 | |a Hecke algebras | |
| 650 | 4 | |a Affine algebraic groups | |
| 650 | 4 | |a Harmonic analysis | |
| 650 | 4 | |a Knizhnik-Zamoldchikov equations | |
| 650 | 4 | |a Orthogonal polynomials | |
| 650 | 0 | 7 | |a Hecke-Algebra |0 (DE-588)4159341-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Affine Algebra |0 (DE-588)4348233-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Harmonische Analyse |0 (DE-588)4023453-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hecke-Algebra |0 (DE-588)4159341-8 |D s |
| 689 | 0 | 1 | |a Affine Algebra |0 (DE-588)4348233-8 |D s |
| 689 | 0 | 2 | |a Harmonische Analyse |0 (DE-588)4023453-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a London Mathematical Society lecture note series |v 319 |w (DE-604)BV000000130 |9 319 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013207873&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-013207873 | |
Datensatz im Suchindex
| DE-19_call_number | 1601/SI 320-319 |
|---|---|
| DE-19_location | 95 |
| DE-BY-TUM_call_number | 0102 MAT 001z 2001 A 985 |
| DE-BY-TUM_katkey | 1515265 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020054037 |
| DE-BY-UBM_katkey | 3164594 |
| DE-BY-UBM_media_number | 41606801370014 |
| _version_ | 1823053089319419904 |
| adam_text | Contents
Preface v
Contents vii
0 Introduction 1
0.0 Universality of Hecke algebras 1
0.0.1 Real and imaginary 1
0.0.2 New vintage 3
0.0.3 Hecke algebras 4
0.1 KZ and Kac Moody algebras 6
0.1.1 Fusion procedure 6
0.1.2 Symmetric spaces 7
0.1.3 KZ and r matrices 8
0.1.4 Integral formulas for KZ 9
0.1.5 From KZ to spherical functions 10
0.2 Double Hecke algebras 11
0.2.1 Missing link? 12
0.2.2 Gauss integrals and sums 14
0.2.3 Difference setup 15
0.2.4 Other directions 16
0.3 DAHA in harmonic analysis 20
0.3.1 Unitary theories 20
0.3.2 From Lie groups to DAHA 22
0.3.3 Elliptic theory 24
0.4 DAHA and Verlinde algebras 27
0.4.1 Abstract Verlinde algebras 27
0.4.2 Operator Verlinde algebras 29
0.4.3 Double Hecke Algebra 30
0.4.4 Nonsymmetric Verlinde algebras 32
0.4.5 Topological interpretation 33
0.5 Applications 35
0.5.1 Flat deformation 35
0.5.2 Rational degeneration 36
0.5.3 Gaussian sums 37
0.5.4 Classification 38
vii
0.5.5 Weyl algebra 39
0.5.6 Diagonal coinvariants 41
1 KZ and QMBP 43
1.0 Soliton connection 43
1.0.1 Classical r matrices 44
1.0.2 Tau function and coinvariant 46
1.0.3 Structure of the chapter 47
1.1 Affine KZ equation 47
1.1.1 Hypergeometric equation 48
1.1.2 AKZ equation of type GL 50
1.1.3 Degenerate affine Hecke algebra 53
1.1.4 Examples 55
1.2 Isomorphism theorems for AKZ 56
1.2.1 Induced representations 57
1.2.2 Monodromy of AKZ 60
1.2.3 Lusztig s isomorphisms 63
1.2.4 AKZ is isomorphic to QMBP 68
1.2.5 The GL case 74
1.3 Isomorphisms for QAKZ 76
1.3.1 Affine Hecke algebras 76
1.3.2 Definition of QAKZ 77
1.3.3 The monodromy cocycle 81
1.3.4 Macdonald s eigenvalue problem 82
1.3.5 Macdonald s operators 88
1.3.6 Arbitrary root systems 90
1.4 DAHA and Macdonald polynomials 92
1.4.1 Rogers polynomials 92
1.4.2 A Hecke algebra approach 94
1.4.3 The GL case 98
1.5 Abstract KZ and elliptic QMBP 105
1.5.1 Abstract r matrices 105
1.5.2 Degenerate DAHA 108
1.5.3 Elliptic QMBP Ill
1.5.4 Double affine KZ 115
1.6 Harish Chandra inversion 116
1.6.1 Affine Weyl groups 118
1.6.2 Degenerate DAHA 119
1.6.3 Differential representation 120
1.6.4 Difference rational case 121
1.6.5 Opdam transform 123
1.6.6 Inverse transform 125
1.7 Factorization and r matrices 128
1.1.1 rsasic trigonometric r matnx iz9
1.7.2 Factorization and r matrices 131
1.7.3 Two conjectures 135
1.7.4 Tau function 136
1.8 Coinvariant, integral formulas 138
1.8.1 Coinvariant 138
1.8.2 Integral formulas . . . . t 141
1.8.3 Proof 144
1.8.4 Comment on KZB 151
2 One dimensional DAHA 154
2.0 Overview 154
2.0.1 Classical origins 154
2.0.2 Main results 155
2.0.3 Other directions 156
2.1 Euler s integral and Gaussian sum 157
2.1.1 Euler s integral, Riemann s zeta 158
2.1.2 Extension by q 159
2.1.3 Mehta Macdonald formula 161
2.1.4 Hankel transform 162
2.1.5 Gaussian sums 163
2.2 Imaginary integration 164
2.2.1 Macdonald s measure 165
2.2.2 Meromorphic continuations 167
2.2.3 Using the constant term 168
2.2.4 Shift operator 171
2.2.5 Applications 173
2.3 Jackson and Gaussian sums 174
2.3.1 Sharp integration 174
2.3.2 Sharp shift formula 177
2.3.3 Roots of unity 178
2.3.4 Gaussian sums 179
2.3.5 Etingof s theorem 181
2.4 Nonsymmetric Hankel transform 184
2.4.1 Operator approach 185
2.4.2 Nonsymmetric theory 187
2.4.3 Rational DAHA 190
2.4.4 Finite dimensional modules 191
2.4.5 Truncated Hankel transform 193
2.5 Polynomial representation 195
2.5.1 Rogers polynomials 195
2.5.2 Nonsymmetric polynomials 197
i 2.5.3 Double affine Hecke algebra 198
{
I
2.5.4 Back to Rogers polynomials .201
2.5.5 Conjugated polynomials • 202
2.6 Four corollaries • 203
2.6.1 Basic definitions 204
2.6.2 Creation operators 205
2.6.3 Standard identities 206
2.6.4 Changing kiok+1 208
2.6.5 Shift formula 209
2.6.6 Proof of the shift formula .210
2.7 DAHA Fourier transforms 212
2.7.1 Functional representation 213
2.7.2 Proof of the master formulas 215
2.7.3 Topological interpretation 216
2.7.4 Plancherel formulas .220
2.7.5 Inverse transforms 224
2.8 Finite dimensional modules 225
2.8.1 Generic q, singular k 226
2.8.2 Additional series .231
2.8.3 Fourier transform .232
2.8.4 Roots of unity q, generic k 234
2.9 Classification, Verlinde algebras 237
2.9.1 The classification list .238
2.9.2 Special spherical representations 240
2.9.3 Perfect representations .246
2.10 Little double Hecke algebra 251
2.10.1 The case of odd N .252
2.10.2 Little double H .253
2.10.3 Half integral it 255
2.10.4 The negative case .257
2.10.5 Deforming Verlinde algebras 259
2.11 DAHA and p adic theory 261
2.11.1 Affine Weyl group .262
2.11.2 Affine Hecke algebra .263
2.11.3 Deforming p adic formulas 265
2.11.4 Fourier transform 267
2.11.5 One dimensional case 268
2.12 Degenerate DAHA 270
2.12.1 Definition of DAHA .271
2.12.2 Polynomials, intertwiners 273
2.12.3 Trigonometric degeneration 274
2.12.4 Rational degeneration 277
2.12.5 Diagonal coinvariants 279
3 General theory 281
3.0 Progenitors 281
3.0.1 Fourier theory 281
3.0.2 Perfect representations 286
3.0.3 Affine Hecke algebras 288
3.0.4 Gauss Selberg integrals and sums 289
3.0.5 From generic q to roots of unity 290
3.0.6 Structure of the chapter 292
3.1 Affine Weyl groups 293
3.1.1 Affine roots 294
3.1.2 Affine length function 296
3.1.3 Reduction modulo W 298
3.1 4 Partial ordering in P 302
3.1.5 Arrows in P 304
3.2 Double H«ke algebras ¦ . 305
3.2.1 Main definition 305
3.2 2 Automorphisms 307
3.2.3 Demazure Lus/,tig operators 310
3.2 4 Filiations 311
3.3 Macdonald polynomials 313
3.3.1 Definitions 314
3.32 Spherical polynomials 317
3.3 3 Intertwining operators 320
3.3.4 Some applications . 323
3.4 Polynomial Fourier transforms 325
3.4.1 Norm formulas •• 325
3.4.2 Discretization 326
3.4.3 Basic transforms . 328
3.4.4 Gauss integrals 331
3.5 .Jackson integrals 334
3.5.1 .Jackson transforms 335
3.5.2 Gauss .Jackson integrals 337
3.5 3 Macdonalds cta i lcntities 339
3.6 Seinisimple representations 342
3.0 1 Eigenvectors and semisimpljeity 343
3.6.2 Main theorem 349
3.6.3 Finite dimensional modules 353
3.0 4 Roots of unity 355
3.0 5 Comment on finite stabilizers 356
3.7 The GL case 357
3.7.1 Generic k 358
3.7.2 Periodic skew diagrams • 360
3.7 3 Partitions 362
3.7.4 Equivalence 364
3.7.5 The classification 365
3.7.6 The column row modules 367
3.7.7 General representations 369
3.8 Spherical representations 371
3.8.1 Spherical and cospherical modules 371
3.8.2 Primitive modules 373
3.8.3 Semisimple spherical modules 376
3.8.4 Spherical modules at roots of unity 378
3.9 Induced and cospherical 382
3.9.1 Notation 382
3.9.2 When are induced cospherical? 384
3.9.3 Irreducible cospherical modules 387
3.9.4 Irreducibility of induced modules 390
3.10 Gaussian and self duality 392
3.10.1 Gaussians 393
3.10.2 Perfect representations 394
3.10.3 Generic q, singular k ¦ ¦ 399
3.10.4 Roots of unity 403
3.11 DAHA and double polynomials . 407
3.11.1 Good reductions 408
3.11.2 Main theorem 409
3.11.3 Weyl algebra 410
3.11.4 Universal DAHA 413
3.11 5 Universal Durikl operators . • 415
3.11.6 Double polynomials 416
Bibliography 418
Index 431
|
| any_adam_object | 1 |
| author | Tscherednik, Iwan Wladimirowitsch 1951- |
| author_GND | (DE-588)1104405687 |
| author_facet | Tscherednik, Iwan Wladimirowitsch 1951- |
| author_role | aut |
| author_sort | Tscherednik, Iwan Wladimirowitsch 1951- |
| author_variant | i w t iw iwt |
| building | Verbundindex |
| bvnumber | BV019883770 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 320 |
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| ctrlnum | (OCoLC)55955646 (DE-599)BVBBV019883770 |
| dewey-full | 512/.22 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.22 |
| dewey-search | 512/.22 |
| dewey-sort | 3512 222 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV019883770 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-03T16:57:40Z |
| institution | BVB |
| isbn | 0521609186 9780521609180 |
| language | English |
| lccn | 2004054567 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013207873 |
| oclc_num | 55955646 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 DE-11 DE-29T |
| owner_facet | DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 DE-11 DE-29T |
| physical | XII, 434 S. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | London Mathematical Society lecture note series |
| series2 | London Mathematical Society lecture note series |
| spellingShingle | Tscherednik, Iwan Wladimirowitsch 1951- Double affine Hecke algebras London Mathematical Society lecture note series Affiene algebra's gtt Analyse harmonique Groupes algébriques affines Harmonische analyse gtt Hecke, Algèbres de Knizhnik-Zamolodchikov, Équations de Polynômes orthogonaux Hecke algebras Affine algebraic groups Harmonic analysis Knizhnik-Zamoldchikov equations Orthogonal polynomials Hecke-Algebra (DE-588)4159341-8 gnd Affine Algebra (DE-588)4348233-8 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4159341-8 (DE-588)4348233-8 (DE-588)4023453-8 |
| title | Double affine Hecke algebras |
| title_auth | Double affine Hecke algebras |
| title_exact_search | Double affine Hecke algebras |
| title_full | Double affine Hecke algebras Ivan Cherednik |
| title_fullStr | Double affine Hecke algebras Ivan Cherednik |
| title_full_unstemmed | Double affine Hecke algebras Ivan Cherednik |
| title_short | Double affine Hecke algebras |
| title_sort | double affine hecke algebras |
| topic | Affiene algebra's gtt Analyse harmonique Groupes algébriques affines Harmonische analyse gtt Hecke, Algèbres de Knizhnik-Zamolodchikov, Équations de Polynômes orthogonaux Hecke algebras Affine algebraic groups Harmonic analysis Knizhnik-Zamoldchikov equations Orthogonal polynomials Hecke-Algebra (DE-588)4159341-8 gnd Affine Algebra (DE-588)4348233-8 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Affiene algebra's Analyse harmonique Groupes algébriques affines Harmonische analyse Hecke, Algèbres de Knizhnik-Zamolodchikov, Équations de Polynômes orthogonaux Hecke algebras Affine algebraic groups Harmonic analysis Knizhnik-Zamoldchikov equations Orthogonal polynomials Hecke-Algebra Affine Algebra Harmonische Analyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013207873&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
| work_keys_str_mv | AT tscherednikiwanwladimirowitsch doubleaffineheckealgebras |