1 Finite Groups of Lie Type .................................... 1
1.1 Algebraic Groups over Finite Fields .................... 1
1.2 Classification Over Finite Fields ...................... 2
1.3 Frobenius Maps ......................................... 3
1.4 Lang Maps .............................................. 4
1.5 Chevalley Groups and Twisted Groups .................... 5
1.6 Example: SL(3, q) and SU(3, q) ......................... 5
1.7 Groups With a BN-Pair .................................. 7
1.8 Notational Conventions ................................. 8
2 Simple Modules ............................................... 9
2.1 Representations and Formal Characters .................. 9
2.2 Simple Modules for Algebraic Groups ................... 10
2.3 Construction of Modules ............................... 11
2.4 Contravariant Forms ................................... 12
2.5 Representations of Frobenius Kernels .................. 13
2.6 Invariants in the Function Algebra .................... 14
2.7 Steinberg's Tensor Product Theorem .................... 15
2.8 Example: SL(2, K) ..................................... 16
2.9 Brauer Theory ......................................... 16
2.10 Counting Semisimple Classes ........................... 16
2.11 Restriction to Finite Subgroups ....................... 17
2.12 Proof of Irreducibility ............................... 17
2.13 Proof of Distinctness: Chevalley Groups ............... 18
2.14 Proof of Distinctness: Twisted Groups ................. 19
2.15 Action of a Sylow p-Subgroup .......................... 20
3 Weyl Modules and Lusztig's Conjecture ....................... 21
3.1 Weyl Modules .......................................... 21
3.2 Restricted Highest Weights ............................ 22
3.3 Cohomology of Line Bundles ............................ 23
3.4 The Affine Weyl Group ................................. 24
3.5 Alcoves ............................................... 25
3.6 Linkage and Translation ............................... 26
3.7 Steinberg Modules ..................................... 26
3.8 Contravariant Form on a Weyl Module ................... 27
3.9 Jantzen Filtration and Sum Formula .................... 28
3.10 Generic Behavior of Weyl Modules ...................... 29
3.11 Lusztig's Conjecture .................................. 30
3.12 Evidence for the Conjecture ........................... 31
4 Computation of Weight Multiplicities ........................ 33
4.1 Weight Spaces in Verma Modules ........................ 33
4.2 Weight Spaces in Characteristic p ..................... 34
4.3 An Easy Example: SL(2, K) ............................. 35
4.4 Computational Algorithms .............................. 35
4.5 Fundamental Modules for Symplectic Groups ............. 37
4.6 Small Weights and Small Characteristics ............... 38
4.7 Small Representations ................................. 38
5 Other Aspects of Simple Modules ............................. 41
5.1 Restriction of Frobenius Maps ......................... 41
5.2 Splitting Fields ...................................... 42
5.3 Special Isogenies ..................................... 43
5.4 Steinberg's Refined Factorization ..................... 44
5.5 Brauer Characters and Grothendieck Rings .............. 45
5.6 Formal Characters and Brauer Characters ............... 46
5.7 Rewriting Formal Sums ................................. 48
5.8 Restricting Highest Weight Modules to Finite
Subgroups ............................................. 48
5.9 Restriction of Weyl Modules ........................... 49
5.10 Restriction of Simple Modules to Levi Subgroups ....... 50
5.11 Restriction to Elementary Abelian p-Subgroups ......... 51
6 Tensor Products ............................................. 53
6.1 Tensor Products of Simple Modules ..................... 53
6.2 Multiplicities ........................................ 54
6.3 Simple Tensor Products ................................ 55
6.4 Semisimple Tensor Products ............................ 56
6.5 Dimensions Divisible by p ............................. 56
6.6 Formal Characters and Multiplicities .................. 57
6.7 Twisting by the Frobenius ............................. 57
6.8 Rewriting Multiplicities .............................. 58
6.9 Multiplicity of the Steinberg Module .................. 59
7 BN-Pairs and Induced Modules ................................ 61
7.1 Weights for Groups with a Split BN-Pair ............... 61
7.2 Principal Series and Intertwining Operators ........... 63
7.3 Examples .............................................. 64
7.4 Summands of Principal Series Modules .................. 64
7.5 Homology Representations .............................. 65
7.6 Comparison with Principal Series ...................... 66
8 Blocks ...................................................... 67
8.1 Blocks of a Group Algebra ............................. 67
8.2 The Defect of a Block ................................. 68
8.3 Groups of Lie Type .................................... 68
8.4 Defect Groups ......................................... 69
8.5 Defect Groups for Groups of Lie Type .................. 70
8.6 Steps in the Proof .................................... 70
8.7 The BN-Pair Setting ................................... 72
8.8 Vertices of Simple Modules ............................ 72
8.9 Representation Type of a Block ........................ 73
9 Projective Modules .......................................... 75
9.1 Projective Modules for Finite Groups .................. 75
9.2 Groups of Lie Type .................................... 76
9.3 The Steinberg Module .................................. 77
9.4 Tensoring with the Steinberg Module ................... 78
9.5 Brauer Characters and Orthogonality Relations ......... 79
9.6 Brauer Characters of PIMs ............................. 79
9.7 A Lower Bound for Dimensions of PIMs .................. 80
9.8 Projective Modules for SL(2, p) ....................... 81
9.9 Brauer Trees for SL(2, p) ............................. 83
9.10 Indecomposable Modules for SL(2, p) ................... 83
9.11 Dimensions of PIMs in Low Ranks ....................... 84
10 Comparison with Frobenius Kernels ........................... 87
10.1 Injective Modules for Frobenius Kernels ............... 87
10.2 Torus Action on Gr -Modules ........................... 89
10.3 Formal Character of Qr(X) ............................. 90
10.4 Lifting PIMs to the Algebraic Group ................... 91
10.5 Some Consequences ..................................... 92
10.6 Tensoring with the Steinberg Module ................... 93
10.7 Small PIMs ............................................ 94
10.8 The Category of G,T-Modules ........................... 94
10.9 Rewriting Multiplicities .............................. 95
10.10 Brauer Characters ..................................... 96
10.11 Statement of the Main Theorem ......................... 96
10.12 Proof of the Main Theorem ............................. 97
10.13 Letting r Grow ........................................ 97
10.14 Some Comparisons in Low Ranks ......................... 99
11 Cartan Invariants .......................................... 101
11.1 Cartan Invariants for Finite Groups .................. 101
11.2 Brauer Characters and Cartan Invariants .............. 102
11.3 Decomposition Numbers ................................ 102
11.4 Groups of Lie Type ................................... 103
11.5 Example: SL(2, p) .................................... 103
11.6 Example: SL(2, q) .................................... 104
11.7 Using Standard Character Data to Compute Cartan
Invariants ........................................... 105
11.8 Conditions for Genericity ............................ 105
11.9 Generic Cartan Invariants ............................ 106
11.10 Growth of Cartan Invariants .......................... 107
11.11 The First Cartan Invariant ........................... 108
11.12 Special Cases ........................................ 108
11.13 Computations of Cartan Matrices ...................... 109
11.14 Example: SL(3, 3) .................................... 110
11.15 Example: Sp(4, 3) and Related Groups ................. 111
11.16 Example: SL(5, 2) .................................... 111
11.17 Conjectures on Block Invariants ...................... 112
12 Extensions of Simple Modules ............................... 115
12.1 The Extension Problem ................................ 115
12.2 Example: SL(2, p) .................................... 116
12.3 The Optimal Situation ................................ 117
12.4 Extensions for Algebraic Groups ...................... 118
12.5 Injectivity Theorem .................................. 119
12.6 Dimensions of Ext Groups ............................. 120
12.7 The Generic Case ..................................... 121
12.8 Truncated Module Categories and Cohomology ........... 121
12.9 Comparison Theorems .................................. 122
12.10 Self-Extensions ...................................... 123
12.11 Special Cases ........................................ 124
12.12 Semisimplicity Criteria .............................. 127
13 Loewy Series ............................................... 129
13.1 Loewy Series ......................................... 129
13.2 Loewy Series for Finite Groups ....................... 130
13.3 Example: SL(2, p) .................................... 130
13.4 Minimal Projective Resolutions ....................... 131
13.5 Example: SL(2, q) .................................... 132
13.6 The Category  ...................................... 132
13.7 Analogies in Characteristic p ........................ 134
13.8 Frobenius Kernels and Algebraic Groups ............... 134
13.9 Example: SL(3, K) .................................... 135
13.10 Principal Series Modules ............................. 136
13.11 Loewy Series for Chevalley Groups .................... 137
13.12 Example: SL(3, 2) .................................... 137
13.13 Example: SL(3, 3) .................................... 138
13.14 Example: SL(4, 2) .................................... 139
13.15 Example: SO(5, 3) .................................... 140
14 Cohomology ................................................. 143
14.1 Cohomology of Finite Groups .......................... 143
14.2 Finite Groups of Lie Type ............................ 144
14.3 Cohomology of Algebraic Groups ....................... 145
14.4 Twisting by Frobenius Maps ........................... 146
14.5 Rational and Generic Cohomology ...................... 146
14.6 Discussion of the Proof .............................. 147
14.7 Example: SL(2, q) .................................... 148
14.8 Explicit Computations ................................ 149
14.9 Recent Developments .................................. 150
15 Complexity and Support Varieties ........................... 151
15.1 Complexity of a Module ............................... 151
15.3 Support Varieties for Restricted Lie Algebras ........ 153
15.4 Support Varieties for Groups of Lie Type ............. 154
15.5 Further Refinements .................................. 155
15.6 Resolutions and Periodicity .......................... 156
15.7 Periodic Modules for Groups of Lie Type .............. 156
16 Ordinary and Modular Representations ....................... 159
16.1 The Decomposition Matrix ............................. 159
16.2 Brauer Characters .................................... 160
16.3 Blocks ............................................... 161
16.4 The Cartan-Brauer Triangle ........................... 161
16.5 Groups of Lie Type ................................... 162
16.6 Blocks of Defect Zero: the Steinberg Character ....... 162
16.7 Cyclic Blocks and Brauer Trees ....................... 163
16.8 Characters of SL(2,q) ................................ 164
16.9 The Brauer Tree of SL(2, p) .......................... 165
16.10 Decomposition Numbers of SL(2, q) .................... 165
16.11 Character Computations for Lie Families .............. 167
16.12 Some Explicit Decomposition Matrices ................. 168
16.13 Special Characters of Sp(2n, q) ...................... 168
16.14 Irreducibility Modulo p .............................. 169
17 Deligne-Lusztig Characters ................................. 171
17.1 Reductive Groups and Frobenius Maps .................. 171
17.2 F-Stable Maximal Tori ................................ 172
17.3 DL Characters ........................................ 173
17.4 Basic Properties of DL Characters .................... 174
17.5 The Decomposition Problem ............................ 175
17.6 PIMs and DL Characters ............................... 176
17.7 DL Characters and Weyl Characters .................... 176
17.8 Generic Decomposition Patterns ....................... 177
17.9 Twisted groups ....................................... 178
17.10 Unipotent Characters ................................. 178
17.11 Semisimple and Regular Characters .................... 179
17.12 Geometric Conjugacy Classes of Cardinality 2 ......... 181
17.13 Jantzen Filtration of a Principal Series Module ...... 181
17.14 Extremal Composition Factors ......................... 182
17.15 Another Look at the Brauer Tree of SL(2, p) .......... 182
17.16 The Brauer Complex ................................... 183
17.17 Dual Formulation ..................................... 184
18 The Groups G2(q) ........................................... 185
18.1 The Groups ........................................... 185
18.2 The Affine Weyl Group ................................ 185
18.3 Weyl Modules and Simple Modules ...................... 186
18.4 Projective Modules for p = 2, 3, 5 ................... 187
18.5 Projective Modules for p ≥ 7 ......................... 191
18.6 Characters of G2{q) .................................. 191
18.7 Reduction Modulo p ................................... 194
18.8 Brauer Complex of G2(5) .............................. 195
19 General and Special Linear Groups .......................... 197
19.1 Representations of GL(n, K) and GL(n,q) .............. 197
19.2 Action on Symmetric Powers ........................... 198
19.3 Dickson Invariants ................................... 198
19.4 Complements .......................................... 199
19.5 Multiplicity of St in Symmetric Powers ............... 200
19.6 Other Simple Modules ................................. 200
19.7 Example: SL(2, p) .................................... 201
19.8 Periodicity for SL(2, q) ............................. 201
19.9 Key Lemma ............................................ 202
19.10 Proof of Periodicity Theorem ......................... 202
19.11 Brauer Lifting ....................................... 204
20 Suzuki and Ree Groups ...................................... 205
20.1 Description of the Groups ............................ 205
20.2 Simple Modules ....................................... 206
20.3 Projective Modules and Blocks ........................ 208
20.4 Cartan Invariants of Suzuki Groups ................... 208
20.5 Cartan Invariants of the Tits Group .................. 209
20.6 Extensions and Cohomology ............................ 210
20.7 Ordinary Characters .................................. 210
20.8 Decomposition Numbers of Suzuki Groups ............... 211
Bibliography Frequently Used Symbols Index
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