BGG范畴O中半单Lie代数的表示 影印版 英文PDF电子书下载

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Chapter 0.Review of Semisimple Lie Algebras 1

0.1.Cartan Decomposition 1

0.2.Root Systems 3

0.3.Weyl Groups 4

0.4.Chevalley-Bruhat Ordering of W 5

0.5.Universal Enveloping Algebras 6

0.6.Integral Weights 7

0.7.Representations 8

0.8.Finite Dimensional Modules 9

0.9.Simple Modules for sl（2，C） 9

Part Ⅰ.Highest Weight Modules 13

Chapter 1.Category O： Basics 13

1.1.Axioms and Consequences 13

1.2.Highest Weight Modules 15

1.3.Verma Modules and Simple Modules 17

1.4.Maximal Vectors in Verma Modules 18

1.5.Example： sl （2，C） 20

1.6.Finite Dimensional Modules 20

1.7.Action of the Center 22

1.8.Central Characters and Linked Weights 24

1.9.Harish-Chandra Homomorphism 25

1.10.Harish-Chandra’s Theorem 26

1.11.Category O is Artinian 28

1.12.Subcategories Ox 30

1.13.Blocks 30

1.14.Formal Characters of Finite Dimensional Modules 32

1.15.Formal Characters of Modules in O 33

1.16.Formal Characters of Verma Modules 34

Notes 35

Chapter 2.Characters of Finite Dimensional Modules 37

2.1.Summary of Prerequisites 37

2.2.Formal Characters Revisited 38

2.3.The Functions p and q 38

2.4.Formulas of Weyl and Kostant 40

2.5.Dimension Formula 42

2.6.Maximal Submodule of M（λ），λ∈ ？＋ 43

2.7.Related Topics 45

Notes 46

Chapter 3.Category O： Methods 47

3.1.Hom and Ext 47

3.2.Duality in O 49

3.3.Duals of Highest Weight Modules 51

3.4.The Reflection Group W[λ] 52

3.5.Dominant and Antidominant Weights 54

3.6.Tensoring Verma Modules with Finite Dimensional Modules 56

3.7.Standard Filtrations 58

3.8.Projectives in O 60

3.9.Indecomposable Projectives 62

3.10.Standard Filtrations of Projectives 64

3.11.BGG Reciprocity 65

3.12.Example： sl（2，C） 66

3.13.Projective Generators and Finite Dimensional Algebras 68

3.14.Contravariant Forms 68

3.15.Universal Construction 70

Notes 71

Chapter 4.Highest Weight Modules Ⅰ 73

4.1.Simple Submodules of Verma Modules 74

4.2.Homomorphisms between Verma Modules 75

4.3.Special Case： Dominant Integral Weights 76

4.4.Simplicity Criterion： Integral Case 77

4.5.Existence of Embeddings： Preliminaries 78

4.6.Existence of Embeddings： Integral Case 79

4.7.Existence of Embeddings： General Case 81

4.8.Simplicity Criterion： General Case 82

4.9.Blocks of O Revisited 83

4.10.Example： Antidominant Projectives 84

4.11.Application to sl（3，C） 85

4.12.Shapovalov Elements 86

4.13.Proof of Shapovalov’s Theorem 88

4.14.A Look Back at Verma’s Thesis 90

Notes 91

Chapter 5.Highest Weight Modules Ⅱ 93

5.1.BGG Theorem 93

5.2.Bruhat Ordering 94

5.3.Jantzen Filtration 95

5.4.Example： sl（3，C） 97

5.5.Application to BGG Theorem 98

5.6.Key Lemma 98

5.7.Proof of Jantzen’s Theorem 100

5.8.Determinant Formula 102

5.9.Details of Shapovalov’s Proof 103

Notes 106

Chapter 6.Extensions and Resolutions 107

6.1.BGG Resolution of a Finite Dimensional Module 108

6.2.Weak BGG Resolution 109

6.3.Exactness of the Sequence 110

6.4.Weights of the Exterior Powers 111

6.5.Extensions of Verma Modules 113

6.6.Application： Bott’s Theorem 115

6.7.Squares 116

6.8.Maps in a BGG Resolution 118

6.9.Homological Dimension 120

6.10.Higher Ext Groups 122

6.11.Vanishing Criteria for Extn 123

6.12.Computation of Ext n O（M（μ），M（λ）v） 124

6.13.Ext Criterion for Standard Filtrations 125

6.14.Characters in Terms of Ext·O 126

6.15.Comparison of Ext·O and Lie Algebra Cohomology 127

Notes 128

Chapter 7.Translation Functors 129

7.1.Translation Functors 130

7.2.Adjoint Functor Property 131

7.3.Weyl Group Geometry 132

7.4.Nonintegral Weights 134

7.5.Key Lemma 135

7.6.Translation Functors and Verma Modules 137

7.7.Translation Functors and Simple Modules 138

7.8.Application： Category Equivalences 138

7.9.Translation to Upper Closures 140

7.10.Character Formulas 142

7.11.Translation Functors and Projective Modules 143

7.12.Translation from a Facet Closure 144

7.13.Example 145

7.14.Translation from a Wall 146

7.15.Wall-Crossing Functors 148

7.16.Self-Dual Projectives 149

Notes 152

Chapter 8.Kazhdan-Lusztig Theory 153

8.1.The Multiplicity Problem for Verma Modules 154

8.2.Hecke Algebras and Kazhdan-Lusztig Polynomials 156

8.3.Examples 157

8.4.Kazhdan-Lusztig Conjecture 159

8.5.Schubert Varieties and KL Polynomials 160

8.6.Example： W of Type C3 161

8.7.Jantzen’s Multiplicity One Criterion 162

8.8.Proof of the KL Conjecture 165

8.9.Outline of the Proof 166

8.10.Ext Functors and Vogan’s Conjecture 168

8.11.KLV Polynomials 169

8.12.The Jantzen Conjecture and the KL Conjecture 171

8.13.Weight Filtrations and Jantzen Filtrations 172

8.14.Review of Loewy Filtrations 173

8.15.Loewy Filtrations and KL Polynomials 174

8.16.Some Details 177

PartⅡ.Further Developments 181

Chapter 9.Parabolic Versions of Category O 181

9.1.Standard Parabolic Subalgebras 182

9.2.Modules for Levi Subalgebras 183

9.3.The Category Op 184

9.4.Parabolic Verma Modules 186

9.5.Example： sl（3，C） 188

9.6.Formal Characters and Composition Factors 189

9.7.Relative Kazhdan-Lusztig Theory 190

9.8.Projectives and BGG Reciprocity in Op 191

9.9.Structure of Parabolic Verma Modules 192

9.10.Maps between Parabolic Verma Modules 193

9.11.Parabolic Verma Modules of Scalar Type 195

9.12.Simplicity of Parabolic Verma Modules 196

9.13.Jantzen’s Simplicity Criterion 198

9.14.Socles and Self-Dual Projectives 199

9.15.Blocks of Op 200

9.16.Analogue of the BGG Resolution 201

9.17.Filtrations and Rigidity 203

9.18.Special Case： Maximal Parabolic Subalgebras 204

Notes 206

Chapter 10.Projective Functors and Principal Series 207

10.1.Functors on Category O 208

10.2.Tensoring With a Dominant Verma Module 210

10.3.Proof of the Theorem 211

10.4.Module Categories 212

10.5.Projective Functors 213

10.6.Annihilator of a Verma Module 215

10.7.Comparison of Hom Spaces 216

10.8.Classification Theorem 218

10.9.Harish-Chandra Modules 219

10.10.Principal Series Modules and Category O 221

Notes 222

Chapter 11.Tilting Modules 223

11.1.Tilting Modules 224

11.2.Indecomposable Tilting Modules 225

11.3.Translation Functors and Tilting Modules 227

11.4.Grothendieck Groups 229

11.5.Subgroups of K 230

11.6.Fusion Rules 231

11.7.Formal Characters 232

11.8.The Parabolic Case 234

Chapter 12.Twisting and Completion Functors 235

12.1.Shuffling Functors 236

12.2.Shuffled Verma Modules 237

12.3.Families of Twisted Verma Modules 239

12.4.Uniqueness of a Family of Twisted Verma Modules 240

12.5.Existence of Twisted Verma Modules 242

12.6.Twisting Functors 242

12.7.Arkhipov’s Construction of Twisting Functors 243

12.8.Twisted Versions of Standard Filtrations 244

12.9.Complete Modules 245

12.10.Enright’s Completions 247

12.11.Completion Functors 248

12.12.Comparison of Functors 249

Chapter 13.Complements 251

13.1.Primitive Ideals in U（g） 252

13.2.Classification of Primitive Ideals 253

13.3.Structure of a Fiber 254

13.4.Kostant’s Problem 255

13.5.Kac-Moody Algebras 256

13.6.Category O for Kac-Moody Algebras 258

13.7.Highest Weight Categories 259

13.8.Blocks and Finite Dimensional Algebras 260

13.9.Quiver Attached to a Block 261

13.10.Representation Type of a Block 263

13.11.Soergel’s Functor V 264

13.12.Coinvariant Algebra of W 265

13.13.Application： Category Equivalence 266

13.14.Endomorphisms and Socles of Projectives 267

13.15.Koszul Duality 268

Bibliography 271

Frequently Used Symbols 283

Index 287

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