Author | Kumar, Shrawan. author |
---|---|
Title | Kac-Moody Groups, their Flag Varieties and Representation Theory [electronic resource] / by Shrawan Kumar |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2002 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0105-2 |
Descript | XV, 609 p. online resource |
I. Kac-Moody Algebras: Basic Theory -- 1. Definition of Kac-Moody Algebras -- 2. Root Space Decomposition -- 3. Weyl Groups Associated to Kac-Moody Algebras -- 4. Dominant Chamber and Tits Cone -- 5. Invariant Bilinear Form and the Casimir Operator -- II. Representation Theory of Kac-Moody Algebras -- 1. Category $$\mathcal{O}$$ -- 2. Weyl-Kac Character Formula -- 3. Shapovalov Bilinear Form -- III. Lie Algebra Homology and Cohomology -- 1. Basic Definitions and Elementary Properties -- 2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky -- 3. Decomposition of the Category $$\mathcal{O}$$ and some Ext Vanishing Results -- 4. Laplacian Calculation -- IV. An Introduction to ind-Varieties and pro-Groups -- 1. Ind-Varieties: Basic Definitions -- 2. Ind-Groups and their Lie Algebras -- 3. Smoothness of ind-Varieties -- 4. An Introduction to pro-Groups and pro-Lie Algebras -- V. Tits Systems: Basic Theory -- 1. An Introduction to Tits Systems -- 2. Refined Tits Systems -- VI. Kac-Moody Groups: Basic Theory -- 1. Definition of Kac-Moody Groups and Parabolic Subgroups -- 2. Representations of Kac-Moody Groups -- VII. Generalized Flag Varieties of Kac-Moody Groups -- 1. Generalized Flag Varieties: Ind-Variety Structure -- 2. Line Bundles on $${\mathcal{X}Ŷ}$$ -- 3. Study of the Group $${\mathcal{U}̂ - }$$ -- 4. Study of the Group $${\mathcal{G}̂{\min }}$$ Defined by Kac-Peterson -- VIII. Demazure and Weyl-Kac Character Formulas -- 1. Cohomology of Certain Line Bundles on $${Z_\mathfrak{w}}$$ -- 2. Normality of Schubert Varieties and the Demazure Character Formula -- 3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem -- IX. BGG and Kempf Resolutions -- 1. BGG Resolution: Algebraic Proof in the Symmetrizable Case -- 2. A Combinatorial Description of the BGG Resolution -- 3. Kempf Resolution -- X. Defining Equations of $$\mathcal{G}/\mathcal{P}$$ and Conjugacy Theorems -- 1. Quadratic Generation of Defining Ideals of $$\mathcal{G}/\mathcal{P}$$ in Projective Embeddings -- 2. Conjugacy Theorems for Lie Algebras -- 3. Conjugacy Theorems for Groups -- XI. Topology of Kac-Moody Groups and Their Flag Varieties -- 1. The Nil-Hecke Ring -- 2. Determination of $$\bar R$$ -- 3. T-equivariant Cohomology of $$\mathcal{G}/\mathcal{P}$$ -- 4. Positivity of the Cup Product in the Cohomology of Flag Varieties -- 5. Degeneracy of the Leray-Serre Spectral Sequence for the Fibration $${\mathcal{G}̂{\min }} \to {\mathcal{G}̂{\min }}/T$$ -- XII. Smoothness and Rational Smoothness of Schubert Varieties -- 1. Singular Locus of Schubert Varieties -- 2. Rational Smoothness of Schubert Varieties -- XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups -- 1. Affine Kac-Moody Lie Algebras -- 2. Affine Kac-Moody Groups -- Appendix A. Results from Algebraic Geometry -- Appendix B. Local Cohomology -- Appendix C. Results from Topology -- Appendix D. Relative Homological Algebra -- Appendix E. An Introduction to Spectral Sequences -- Index of Notation