Volume: 94; 2008; 289 pp; Softcover
MSC: Primary 17; Secondary 20; 22
Print ISBN: 978-1-4704-6326-7
Product Code: GSM/94.S
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Supplemental Materials
Representations of Semisimple Lie Algebras in the BGG Category \(\mathscr {O}\)
Share this pageJames E. Humphreys
This is the first textbook treatment of work leading to the landmark
1979 Kazhdan–Lusztig Conjecture on characters of simple highest
weight modules for a semisimple Lie algebra \(\mathfrak{g}\)
over \(\mathbb {C}\). The setting is the module category
\(\mathscr {O}\) introduced by Bernstein–Gelfand–Gelfand,
which includes all highest weight modules for \(\mathfrak{g}\) such as
Verma modules and finite dimensional simple modules. Analogues of this
category have become influential in many areas of representation
theory.
Part I can be used as a text for independent study or for a
mid-level one semester graduate course; it includes exercises and
examples. The main prerequisite is familiarity with the structure
theory of \(\mathfrak{g}\). Basic techniques in category
\(\mathscr {O}\) such as BGG Reciprocity and Jantzen's
translation functors are developed, culminating in an overview of the
proof of the Kazhdan–Lusztig Conjecture (due to
Beilinson–Bernstein and Brylinski–Kashiwara). The full proof
however is beyond the scope of this book, requiring deep geometric
methods: \(D\)-modules and perverse sheaves on the flag
variety. Part II introduces closely related topics important in
current research: parabolic category \(\mathscr {O}\),
projective functors, tilting modules, twisting and completion
functors, and Koszul duality theorem of
Beilinson–Ginzburg–Soergel.
Readership
Graduate students and research mathematicians interested in Lie theory, and representation theory.
Reviews & Endorsements
One of the goals Humphreys had in mind was to provide a textbook suitable for graduate students. This has been achieved by keeping prerequisites to a minimum, by careful dealing with technical parts of the proofs, and by offering a large number of exercises.
-- Mathematical Reviews
Table of Contents
Table of Contents
Representations of Semisimple Lie Algebras in the BGG Category $\mathscr{O}$
Table of Contents pages: 1 2
- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xv16 free
- Chapter 0. Review of Semisimple Lie Algebras 118 free
- Part I. Highest Weight Modules 1128
- Chapter 1. Category O: Basics 1330
- §1.1. Axioms and Consequences 1330
- §1.2. Highest Weight Modules 1532
- §1.3. Verma Modules and Simple Modules 1734
- §1.4. Maximal Vectors in Verma Modules 1835
- §1.5. Example: sl(2, C) 2037
- §1.6. Finite Dimensional Modules 2037
- §1.7. Action of the Center 2239
- §1.8. Central Characters and Linked Weights 2441
- §1.9. Harish-Chandra Homomorphism 2542
- §1.10. Harish-Chandra's Theorem 2643
- §1.11. Category O is Artinian 2845
- §1.12. Subcategories O[sub(x)] 3047
- §1.13. Blocks 3047
- §1.14. Formal Characters of Finite Dimensional Modules 3249
- §1.15. Formal Characters of Modules in O 3350
- §1.16. Formal Characters of Verma Modules 3451
- Notes 3552
- Chapter 2. Characters of Finite Dimensional Modules 3754
- Chapter 3. Category O: Methods 4764
- §3.1. Horn and Ext 4764
- §3.2. Duality in O 4966
- §3.3. Duals of Highest Weight Modules 5168
- §3.4. The Reflection Group M[sub(λ)] 5269
- §3.5. Dominant and Antidominant Weights 5471
- §3.6. Tensoring Verma Modules with Finite Dimensional Modules 5673
- §3.7. Standard Filtrations 5875
- §3.8. Projectives in O 6077
- §3.9. Indecomposable Projectives 6279
- §3.10. Standard Filtrations of Projectives 6481
- §3.11. BGG Reciprocity 6582
- §3.12. Example: sI(2, C) 6683
- §3.13. Projective Generators and Finite Dimensional Algebras 6885
- §3.14. Contravariant Forms 6885
- §3.15. Universal Construction 7087
- Notes 7188
- Chapter 4. Highest Weight Modules I 7390
- §4.1. Simple Submodules of Verma Modules 7491
- §4.2. Homomorphisms between Verma Modules 7592
- §4.3. Special Case: Dominant Integral Weights 7693
- §4.4. Simplicity Criterion: Integral Case 7794
- §4.5. Existence of Embeddings: Preliminaries 7895
- §4.6. Existence of Embeddings: Integral Case 7996
- §4.7. Existence of Embeddings: General Case 8198
- §4.8. Simplicity Criterion: General Case 8299
- §4.9. Blocks of O Revisited 83100
- §4.10. Example: Antidominant Projectives 84101
- §4.11. Application to sl(3, C) 85102
- §4.12. Shapovalov Elements 86103
- §4.13. Proof of Shapovalov's Theorem 88105
- §4.14. A Look Back at Verma's Thesis 90107
- Notes 91108
- Chapter 5. Highest Weight Modules II 93110
- §5.1. BGG Theorem 93110
- §5.2. Bruhat Ordering 94111
- §5.3. Jantzen Filtration 95112
- §5.4. Example: sl(3, C) 97114
- §5.5. Application to BGG Theorem 98115
- §5.6. Key Lemma 98115
- §5.7. Proof of Jantzen's Theorem 100117
- §5.8. Determinant Formula 102119
- §5.9. Details of Shapovalov's Proof 103120
- Notes 106123
- Chapter 6. Extensions and Resolutions 107124
- §6.1. BGG Resolution of a Finite Dimensional Module 108125
- §6.2. Weak BGG Resolution 109126
- §6.3. Exactness of the Sequence 110127
- §6.4. Weights of the Exterior Powers 111128
- 56.5. Extensions of Verma Modules 113130
- §6.6. Application: Bott's Theorem 115132
- §6.7. Squares 116133
- §6.8. Maps in a BGG Resolution 118135
- §6.9. Homological Dimension 120137
- §6.10. Higher Ext Groups 122139
- §6.11. Vanishing Criteria for Ext[sup(n)] 123140
- §6.12. Computation of Ext[sup(n)][sub(O)](M(μ), M(λ)[sup(∨)]) 124141
- §6.13. Ext Criterion for Standard Filtrations 125142
- §6.14. Characters in Terms of Ext[sup(⋅)][sub(O)] 126143
- §6.15. Comparison of Ext[sup(⋅)][sub(O)] and Lie Algebra Cohomology 127144
- Notes 128145
- Chapter 7. Translation Functors 129146
- §7.1. Translation Functors 130147
- §7.2. Adjoint Functor Property 131148
- §7.3. Weyl Group Geometry 132149
- §7.4. Nonintegral Weights 134151
- §7.5. Key Lemma 135152
- §7.6. Translation Functors and Verma Modules 137154
- §7.7. Translation Functors and Simple Modules 138155
- §7.8. Application: Category Equivalences 138155
- §7.9. Translation to Upper Closures 140157
- §7.10. Character Formulas 142159
- §7.11. Translation Functors and Projective Modules 143160
- §7.12. Translation from a Facet Closure 144161
- §7.13. Example 145162
- §7.14. Translation from a Wall 146163
- §7.15. Wall-Crossing Functors 148165
- §7.16. Self-Dual Projectives 149166
- Notes 152169
- Chapter 8. Kazhdan-Lusztig Theory 153170
- §8.1. The Multiplicity Problem for Verma Modules 154171
- §8.2. Hecke Algebras and Kazhdan–Lusztig Polynomials 156173
- §8.3. Examples 157174
- §8.4. Kazhdan-Lusztig Conjecture 159176
- §8.5. Schubert Varieties and KL Polynomials 160177
- §8.6. Example: W of Type C[sub(3)] 161178
- §8.7. Jantzen's Multiplicity One Criterion 162179
- §8.8. Proof of the KL Conjecture 165182
- §8.9. Outline of the Proof 166183
- §8.10. Ext Functors and Vogan's Conjecture 168185
- §8.11. KLV Polynomials 169186
- §8.12. The Jantzen Conjecture and the KL Conjecture 171188
- §8.13. Weight Filtrations and Jantzen Filtrations 172189
- §8.14. Review of Loewy Filtrations 173190
- §8.15. Loewy Filtrations and KL Polynomials 174191
- §8.16. Some Details 177194
- Part II. Further Developments 179196
- Chapter 9. Parabolic Versions of Category O 181198
- §9.1. Standard Parabolic Subalgebras 182199
- §9.2. Modules for Levi Subalgebras 183200
- §9.3. The Category O[sup(p)] 184201
- §9.4. Parabolic Verma Modules 186203
- §9.5. Example: sI(3, C) 188205
- §9.6. Formal Characters and Composition Factors 189206
- §9.7. Relative Kazhdan-Lusztig Theory 190207
- §9.8. Projectives and BGG Reciprocity in O[sup(p)] 191208
- §9.9. Structure of Parabolic Verma Modules 192209
- §9.10. Maps between Parabolic Verma Modules 193210
- §9.11. Parabolic Verma Modules of Scalar Type 195212
- §9.12. Simplicity of Parabolic Verma Modules 196213
- §9.13. Jantzen's Simplicity Criterion 198215
- §9.14. Socles and Self-Dual Projectives 199216
- §9.15. Blocks of O[sup(p)] 200217
- §9.16. Analogue of the BGG Resolution 201218
- §9.17. Filtrations and Rigidity 203220
- §9.18. Special Case: Maximal Parabolic Subalgebras 204221
- Notes 206223
- Chapter 10. Projective Functors and Principal Series 207224
- §10.1. Functors on Category O 208225
- §10.2. Tensoring With a Dominant Verma Module 210227
- §10.3. Proof of the Theorem 211228
- §10.4. Module Categories 212229
- §10.5. Projective Functors 213230
- §10.6. Annihilator of a Verma Module 215232
- §10.7. Comparison of Horn Spaces 216233
- §10.8. Classification Theorem 218235
- §10.9. Harish-Chandra Modules 219236
- §10.10. Principal Series Modules and Category O 221238
- Notes 222239
- Chapter 11. Tilting Modules 223240
- Chapter 12. Twisting and Completion Functors 235252
- §12.1. Shuffling Functors 236253
- §12.2. Shuffled Verma Modules 237254
- §12.3. Families of Twisted Verma Modules 239256
- §12.4. Uniqueness of a Family of Twisted Verma Modules 240257
- §12.5. Existence of Twisted Verma Modules 242259
- §12.6, Twisting Functors 242259
- §12.7. Arkhipov's Construction of Twisting Functors 243260
- §12.8. Twisted Versions of Standard Filtrations 244261
- §12.9. Complete Modules 245262
- §12.10. Enright's Completions 247264
- §12.11. Completion Functors 248265
- §12.12. Comparison of Functors 249266
- Chapter 13. Complements 251268
Table of Contents pages: 1 2