Kac-Moody Groups, their Flag Varieties and Representation Theory

Progress in Mathematics 204......Page 1
Kac-Moody Groups, their Flag Varieties and Representation Theory......Page 2
Contents......Page 5
Preface......Page 9
Convention......Page 12
I. Kac-Moody Algebras: Basic Theory......Page 15
1.1. Definition of Kac-Moody Algebras......Page 16
1.2. Root Space Decomposition......Page 19
1.3. Weyl Groups Associated to Kac-Moody Algebras......Page 25
1.4. Dominant Chamber and Tits Cone......Page 41
1.5. Invariant Bilinear Form and the Casimir Operator......Page 43
II. Representation Theory of Kac-Moody Algebras......Page 53
2.1. Category O......Page 54
2.2. Weyl-Kac Character Formula......Page 61
2.3. Shapavalov Bilinear Form......Page 65
III. Lie Algebra Homology and Cohomology......Page 81
3.1. Basic Definitions and Elementary Properties......Page 82
3.2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky......Page 94
3.3. Decomposition of the Category O and some Ext Vanishing Results......Page 104
3.4. Laplacian Calculation......Page 111
IV. An Introduction to ind-Varieties and pro-Groups......Page 122
4.1. Ind-Varieties: Basic Definitions......Page 123
4.2. Ind-Groups and their Lie Algebras......Page 127
4.3. Smoothness of ind-Varieties......Page 135
4.4. An Introduction to pro-Groups and pro-Lie Algebras......Page 142
5.1. An Introduction to Tits Systems......Page 162
5.2. Refined Tits Systems......Page 179
VI. Kac-Moody Groups: Basic Theory......Page 186
6.1. Definition of Kac-Moody Groups and Parabolic Subgroups......Page 187
6.2. Representations of Kac-Moody Groups......Page 200
VII. Generalized Flag Varieties of Kac-Moody Groups......Page 212
7.1. Generalized Flag Varieties„Ind-Variety Structure......Page 214
7.2. Line Bundles on XY......Page 232
7.3. Study of the group U......Page 234
7.4. Study of the Group Gmin Defined by Kac-Peterson......Page 241
VIII. Demazure and Weyl-Kac Character Formulas......Page 258
8.1. Cohomology of Certain Line Bundles on Zw......Page 259
8.2. Normality of Schubert Varieties and the Demazure Character Formula......Page 286
8.3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem......Page 294
IX. BGG and Kempf Resolutions......Page 308
9.1. BGG Resolution: An Algebraic Proof in the Symmetrizable Case......Page 310
9.2. A Combinatorial Description of the BGG Resolution......Page 318
9.3. Kempf Resolution......Page 334
X. Defining Equations of G/P and Conjugacy Theorems......Page 350
10.1. Quadratic Generation of Defining Ideals of G/P in Projective Embeddings......Page 352
10.2. Conjugacy Theorems for Lie Algebras......Page 360
10.3. Conjugacy Theorems for Groups......Page 371
XI. Topology of Kac-Moody Groups and Their Flag Varieties......Page 382
11.1. The Nil-Hecke Ring......Page 384
11.2. Determination of R......Page 405
11.3. T-equivariant Cohomology of G/B......Page 409
11.4. Positivity of the Cup Product in the Cohomology of Flag Varieties......Page 429
þÿ......Page 440
XII. Smoothness and Rational Smoothness of Schubert Varieties......Page 460
12.1. Singular Locus of Schubert Varieties......Page 462
12.2. Rational Smoothness of Schubert Varieties......Page 478
XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups......Page 494
13.1. Affine Kac-Moody Lie Algebras......Page 495
13.2. Affine Kac-Moody Groups......Page 503
Appendix A. Results from Algebraic Geometry......Page 524
Appendix B. Local Cohomology......Page 540
Appendix C. Results from Topology......Page 546
Appendix D. Homological Algebra......Page 552
Appendix E. An Introduction to Spectral Sequences......Page 562
Bibliography......Page 572
Index of Notation......Page 604
Index......Page 610