Representation theory of finite reductive groups

Half-title......Page 2
Title......Page 4
Copyright......Page 5
Table of Contents......Page 6
Preface......Page 12
Terminology......Page 16
PART I: Representing finite BN-pairs......Page 20
1 Cuspidality in finite groups......Page 22
1.1. Subquotients and associated restrictions......Page 23
1.2. Cuspidality and induction......Page 25
1.3. Morphisms and an invariance theorem......Page 27
1.4. Endomorphism algebras of induced cuspidal modules......Page 30
1.5. Self-injective endomorphism rings and an equivalence of categories......Page 33
1.6. Structure of induced cuspidal modules and series......Page 36
Notes......Page 39
2 Finite BN-pairs......Page 41
2.1. Coxeter groups and root systems......Page 42
2.2. BN-pairs......Page 46
2.3. Root subgroups......Page 48
2.4. Levi decompositions......Page 51
2.5. Other properties of split BN-pairs......Page 54
Notes......Page 59
3 Modular Hecke algebras for finite BN-pairs......Page 60
3.1. Hecke algebras in transversal characteristics......Page 61
3.2. Quotient root system and a presentation of the Hecke algebra......Page 66
Notes......Page 73
4 The modular duality functor and derived category......Page 74
4.1.1. Complexes and associated categories......Page 75
4.1.2. Simplicial schemes......Page 76
4.1.4. Associated homology complexes......Page 77
4.2. Fixed point coefficient system and cuspidality......Page 78
4.3. The case of finite BN-pairs......Page 82
4.4. Duality functor as a derived equivalence......Page 86
4.5. A theorem of Curtis type......Page 88
Notes......Page 91
5 Local methods for the transversal characteristics......Page 93
5.1. Local methods and two main theorems of Brauer's......Page 94
5.2. A model: blocks of symmetric groups......Page 97
5.3. Principal series and the principal block......Page 101
5.4. Hecke algebras and decomposition matrices......Page 103
5.5. A proof of Brauer's third Main Theorem......Page 105
Notes......Page 106
6.1. Modular Hecke algebra associated with a Sylow ρ-subgroup......Page 107
6.2. Some modules in characteristic ρ......Page 112
6.3. Alperin’s weight conjecture in characteristic ρ......Page 114
6.4. The ρ-blocks......Page 116
Notes......Page 119
PART II: Deligne–Lusztig varieties, rational series, and Morita equivalences......Page 120
7 Finite reductive groups and Deligne–Lusztig varieties......Page 122
7.1. Reductive groups and Lang's theorem......Page 123
7.2. Varieties defined by the Lang map......Page 124
7.3. Deligne–Lusztig varieties......Page 128
7.4. Deligne–Lusztig varieties are quasi-affine......Page 133
Notes......Page 136
8 Characters of finite reductive groups......Page 137
8.1. Reductive groups, isogenies......Page 138
8.2. Some exact sequences and groups in duality......Page 141
8.3. Twisted induction......Page 144
8.4. Lusztig's series......Page 146
Notes......Page 149
9 Blocks of finite reductive groups and rational series......Page 150
9.1. Blocks and characters......Page 151
9.2. Blocks and rational series......Page 152
9.3. Morita equivalence and ordinary characters......Page 155
Notes......Page 159
10 Jordan decomposition as a Morita equivalence: the main reductions......Page 160
10.1. The condition…......Page 161
10.2. A first reduction......Page 163
10.3. More notation: smooth compactifications......Page 165
10.4. Ramification and generation......Page 168
10.5. A second reduction......Page 169
Notes......Page 173
11 Jordan decomposition as a Morita equivalence: sheaves......Page 174
11.1. Ramification in Deligne–Lusztig varieties......Page 175
11.2. Coroot lattices associated with intervals......Page 181
11.3. Deligne–Lusztig varieties associated with intervals......Page 184
11.4. Application: some mapping cones......Page 187
Notes......Page 191
12 Jordan decomposition as a Morita equivalence: modules......Page 192
12.1. Generating perfect complexes......Page 193
12.2. The case of modules induced by Deligne–Lusztig varieties......Page 195
12.3. Varieties of minimal dimension inducing a simple module......Page 196
12.4. Disjunction of series......Page 200
Notes......Page 205
PART III: Unipotent characters and unipotent blocks......Page 206
13.1. Polynomial orders of F-stable tori......Page 208
13.2. Good primes......Page 212
13.3. Centralizers of l-subgroups and some Levi subgroups......Page 213
Notes......Page 217
14.1. Dual conjugacy classes for l-elements......Page 218
14.2. Basic sets in the case of connected center......Page 220
Notes......Page 223
15 Jordan decomposition of characters......Page 224
15.1. From non-connected center to connected center and dual morphism......Page 225
15.2. Jordan decomposition of characters......Page 228
Notes......Page 237
16 On conjugacy classes in type D......Page 238
16.1. Notation; some power series......Page 239
16.2. Orthogonal groups......Page 240
16.3. Special orthogonal groups and their derived subgroup; Clifford groups......Page 246
16.4. Spin2n(F)......Page 254
16.5. Non-semi-simple groups, conformal groups......Page 258
16.6. Group with connected center and derived group Spin2n(F); conjugacy classes......Page 264
16.7. Group with connected center and derived group Spin2n(F); Jordan decomposition of characters......Page 267
16.8. Last computation......Page 269
Notes......Page 277
17 Standard isomorphisms for unipotent blocks......Page 278
17.1. The set of unipotent blocks......Page 279
17.2. l-series and non-connected center......Page 280
17.3. A ring isomorphism......Page 283
Note......Page 286
PART IV: Decomposition numbers and q-Schur algebras......Page 288
18 Some integral Hecke algebras......Page 290
18.1. Hecke algebras and sign ideals......Page 291
18.2. Hecke algebras of type A......Page 294
18.3. Hecke algebras of type BC; Hoefsmit’s matrices and Jucys–Murphy elements......Page 298
18.4. Hecke algebras of type BC: some computations......Page 300
18.5. Hecke algebras of type BC: a Morita equivalence......Page 304
18.6. Cyclic Clifford theory and decomposition numbers......Page 307
Notes......Page 314
19 Decomposition numbers and q-Schur algebras: general linear groups......Page 316
19.1. Hom functors and decomposition numbers......Page 317
19.2. Cuspidal simple modules and Gelfand–Graev lattices......Page 320
19.3. Simple modules and decomposition matrices for unipotent blocks......Page 324
19.4. Modular Harish-Chandra series......Page 328
Notes......Page 336
20 Decomposition numbers and Fei-Schur algebras: linear primes......Page 337
20.1. Finite classical groups and linear primes......Page 338
20.2. Hecke algebras......Page 341
20.3. Type BC......Page 345
20.4. Type D......Page 347
Note......Page 349
PART V: Unipotent blocks and twisted induction......Page 350
21.1. "Connected" subpairs in finite reductive groups......Page 352
21.2. Twisted induction for blocks......Page 353
21.3. A bad prime......Page 360
Notes......Page 362
22 Unipotent blocks and generalized Harish-Chandra theory......Page 364
22.1. Local subgroups in finite reductive groups, l-elements and tori......Page 365
22.2. The theorem......Page 369
22.3. Self-centralizing subpairs......Page 371
22.4. The defect groups......Page 373
Notes......Page 377
23 Local structure and ring structure of unipotent blocks......Page 379
23.1. Non-unipotent characters in unipotent blocks......Page 380
23.2. Control subgroups......Page 382
23.3. (q – 1)-blocks and abelian defect conjecture......Page 385
Notes......Page 390
Appendices......Page 392
A1.1. Abelian categories......Page 393
A1.3. The mapping cone......Page 394
A1.5. The homotopic category......Page 395
A1.6. Derived categories......Page 396
A1.7. Cones and distinguished triangles......Page 397
A1.9. Composition of derived functors......Page 398
A1.11. Bi-functors......Page 399
A1.12. Module categories......Page 400
A1.13. Sheaves on topological spaces......Page 401
A1.14. Locally constant sheaves and the fundamental group......Page 403
A1.15. Derived operations on sheaves......Page 404
Notes......Page 406
A2.1. Affine F-varieties......Page 408
A2.2. Locally ringed spaces and F-varieties......Page 409
A2.3. Tangent sheaf, smoothness......Page 411
A2.4. Linear algebraic groups and reductive groups......Page 412
A2.6. Morphisms and quotients......Page 414
A2.7. Schemes......Page 416
A2.8. Coherent sheaves......Page 418
A2.9. Vector bundles......Page 419
A2.10. A criterion of quasi-affinity......Page 420
Notes......Page 422
A3.1. The étale topology......Page 423
A3.2. Sheaves for the étale topology......Page 424
A3.3. Basic operations on sheaves......Page 425
A3.4. Homology and derived functors......Page 426
A3.6. Homology and direct images with compact support......Page 427
A3.8. Coefficients......Page 428
A3.9. The "open-closed" situation......Page 429
A3.11. Projection and Künneth formulae......Page 430
A3.12. Poincaré–Verdier duality and twisted inverse images......Page 431
A3.13. Purity......Page 432
A3.15. Finite group actions and projectivity......Page 433
A3.16. Locally constant sheaves and the fundamental group......Page 434
A3.17. Tame ramification along a divisor with normal crossings......Page 436
A3.18. Tame ramification and direct images......Page 437
Notes......Page 440
References......Page 441
Index......Page 450