Kac-Moody Groups, their Flag Varieties and Representation Theory

Shrawan Kumar

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 cones
5. Invariant bilinear form and the Casimir operator

II. Representation theory of Kac-Moody algebras

1. Category 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 O and some Ext vanishing results
4. Laplacian calculation

IV. An introdunction 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. Lein bundles on X^Y
3. Study of the group U^-
4. Study of the group G^min defined by Kac-Peterson

VIII. Demazure and Weyl-Kac character formulas

1. Cohomology of certain line bundles on Z_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 resolution

1. BGG resolution: algebraic proof in the symmetrizable case
2. A combinatorial description of the BGG resolution
3. Kempf resolution

X. Defining equations of G/P and conjugacy theorems

1. Quadratic generation of defining ideals of G/P in projective embeddings
2. Conjugacy theorems for Lie algebras
3. Conjugacy theorems for groups

XI. Topology of Kac-Moody groups and theor flag varieties

1. The nil-Hecke ring
2. Determination of R-bar
3. T-equivariant cohomology of G/B
4. Positivity of the cup product in the cohomology of flag varieties
5. Degeneracy of the Laray-Serre spectral sequence for the fibration G^min over 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

A. Results from algebraic geometry

B. Local cohomology

C. Results from topology

D. Relative homological algebra

E. An introduction to spectral sequences