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.

Inhalt

Introduction Kac--Moody Algebras -- Basic Theory Representation Theory of Kac--Moody Algebras Lie Algebra Homology and Cohomology An Introduction to ind-Varieties and pro-Groups Tits Systems -- Basic Theory Kac--Moody Groups -- Basic Theory Generalized Flag Varieties of Kac--Moody Groups Demazure and Weyl--Kac Character Formulas BGG and Kempf Resolutions Defining Equations of G/P and Conjugacy Theorems Topology of Kac-Moody Groups and Their Flag Varieties Smoothness and Rational Smoothness of Schubert Varieties An Introduction to Affine Kac-Moody Lie Algebras and 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 Bibliography Index of Notation * Index