Kac-Moody Groups, their Flag Varieties and Representation Theory

Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge­ bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan­ dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g.

Table of Contents:
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 $${\mathcal{X}^Y}$$.- 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.