Representations of Semisimple Lie Algebras in the BGG Category 0

Transcription

1 Representations of Semisimple Lie Algebras in the BGG Category 0 James E. Humphreys Graduate Studies in Mathematics Volume 94 American Mathematical Society Providence, Rhode Island

2 Contents Preface Chapter 0. Review of Semisimple Lie Algebras Cartan Decomposition Root Systems Weyl Groups Chevalley-Bruhat Ordering of W Universal Enveloping Algebras Integral Weights Representations Finite Dimensional Modules Simple Modules for sl(2, C) 9 Part I. Highest Weight Modules Chapter 1. Category O: Basics Axioms and Consequences Highest Weight Modules Verma Modules and Simple Modules Maximal Vectors in Verma Modules Example: sl(2,c) Finite Dimensional Modules Action of the Center Central Characters and Linked Weights 24 xv vii

3 viii Contents 1.9. Harish-Chandra Homomorphism Harish-Chandra's Theorem Category O is Artinian Subcategories O x Blocks Formal Characters of Finite Dimensional Modules Formal Characters of Modules in O Formal Characters of Verma Modules 34 Notes 35 Chapter 2. Characters of Finite Dimensional Modules Summary of Prerequisites Formal Characters Revisited The Functions p and q Formulas of Weyl and Kostant Dimension Formula Maximal Submodule of M(X), A G A Related Topics 45 Notes - 46 Chapter 3. Category O: Methods Horn and Ext Duality in O Duals of Highest Weight Modules The Reflection Group W [X ] Dominant and Antidominant Weights Tensoring Verma Modules with Finite Dimensional Modules Standard Filtrations Projectives in O Indecomposable Projectives Standard Filtrations of Projectives BGG Reciprocity Example: sl(2, C) Projective Generators and Finite Dimensional Algebras Contravariant Forms Universal Construction 70

4 Contents ix Notes 71 Chapter 4. Highest Weight Modules I Simple Submodules of Verma Modules Homomorphisms between Verma Modules Special Case: Dominant Integral Weights Simplicity Criterion: Integral Case Existence of Embeddings: Preliminaries Existence of Embeddings: Integral Case Existence of Embeddings: General Case Simplicity Criterion: General Case Blocks of O Revisited Example: Antidominant Projectives Application to sl(3, C) Shapovalov Elements Proof of Shapovalov's Theorem A Look Back at Verma's Thesis 90 Notes 91 Chapter 5. Highest Weight Modules II BGG Theorem Bruhat Ordering Jantzen Filtration Example: sl(3, C) Application to BGG Theorem Key Lemma Proof of Jantzen's Theorem Determinant Formula Details of Shapovalov's Proof 103 Notes 106 Chapter 6. Extensions and Resolutions BGG Resolution of a Finite Dimensional Module Weak BGG Resolution Exactness of the Sequence Weights of the Exterior Powers Extensions of Verma Modules 113

5 x Contents 6.6. Application: Bott's Theorem Squares Maps in a BGG Resolution Homological Dimension Higher Ext Groups 122 ^ Vanishing Criteria for Ext" Computation of Extg,(M(»,M(A) v ) Ext Criterion for Standard Filtrations Characters in Terms of Exf o Comparison of Ext^ and Lie Algebra Cohomology 127 Notes 128 Chapter 7. Translation Functors Translation Functors Adjoint Functor Property Weyl Group Geometry Nonintegral Weights Key Lemma Translation Functors and Verma Modules Translation Functors and Simple Modules Application: Category Equivalences Translation to Upper Closures Character Formulas Translation Functors and Projective Modules Translation from a Facet Closure Example Translation from a Wall Wall-Crossing Functors Self-Dual Projectives 149 Notes 152 Chapter 8. Kazhdan-Lusztig Theory The Multiplicity Problem for Verma Modules Hecke Algebras and Kazhdan-Lusztig Polynomials Examples Kazhdan-Lusztig Conjecture 159

6 Contents xi 8.5. Schubert Varieties and KL Polynomials Example: W of Type C Jantzen's Multiplicity One Criterion Proof of the KL Conjecture Outline of the Proof Ext Functors and Vogan's Conjecture KLV Polynomials The Jantzen Conjecture and the KL Conjecture Weight Filtrations and Jantzen Filtrations Review of Loewy Filtrations Loewy Filtrations and KL Polynomials Some Details 177 Part II. Further Developments Chapter 9. Parabolic Versions of Category O Standard Parabolic Subalgebras Modules for Levi Subalgebras The Category OP Parabolic Verma Modules Example: sl(3, C) Formal Characters and Composition Factors Relative Kazhdan-Lusztig Theory Projectives and BGG Reciprocity in O p Structure of Parabolic Verma Modules Maps between Parabolic Verma Modules Parabolic Verma Modules of Scalar Type Simplicity of Parabolic Verma Modules Jantzen's Simplicity Criterion Socles and Self-Dual Projectives Blocks of Of Analogue of the BGG Resolution Filtrations and Rigidity Special Case: Maximal Parabolic Subalgebras 204 Notes 206 Chapter 10. Projective Functors and Principal Series 207

7 xii Contents Functors on Category O Tensoring With a Dominant Verma Module Proof of the Theorem Module Categories Projective Functors Annihilator of a Verma Module Comparison of Horn Spaces Classification Theorem Harish-Chandra Modules Principal Series Modules and Category O 221 Notes 222 Chapter 11. Tilting Modules Tilting Modules Indecomposable Tilting Modules Translation Functors and Tilting Modules Grothendieck Groups Subgroups of fc Fusion Rules Formal Characters The Parabolic Case 234 Chapter 12. Twisting and Completion Functors Shuffling Functors Shuffled Verma Modules Families of Twisted Verma Modules Uniqueness of a Family of Twisted Verma Modules Existence of Twisted Verma Modules Twisting Functors Arkhipov's Construction of Twisting Functors Twisted Versions of Standard Filtrations Complete Modules Enright's Completions Completion Functors Comparison of Functors 249 Chapter 13. Complements 251

8 Contents xiii Primitive Ideals in U(g) Classification of Primitive Ideals Structure of a Fiber Kostant's Problem Kac-Moody Algebras Category O for Kac-Moody Algebras Highest Weight Categories Blocks and Finite Dimensional Algebras Quiver Attached to a Block Representation Type of a Block Soergel's Functor V Coinvariant Algebra of W Application: Category Equivalence Endomorphisms and Socles of Projectives Koszul Duality 268 Bibliography 271 Frequently Used Symbols 283 Index 287