Kac-Moody Groups, Their Flag Varieties, and Representation Theory (Freiburger Veroffentlichungen Zum Religionsrecht,)

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. I. Kac-moody Algebras: Basic Theory -- Ii. Representation Theory Of Kac-moody Algebras -- Iii. Lie Algebra Homology And Cohomology -- Iv. An Introduction To Ind-varieties And Pro-groups -- V. Tits Systems: Basic Theory -- Vi. Kac-moody Groups: Basic Theory -- Vii. Generalized Flag Varieties Of Kac-moody Groups -- Viii. Demazure And Weyl-kac Character Formulas -- Ix. Bgg And Kempf Resolutions -- X. Defining Equations Of G/p And Conjugacy Theorems -- Xi. Topology Of Kac-moody Groups And Their Flag Varieties -- Xii. Smoothness And Rational Smoothness Of Schubert Varieties -- Xiii. An Introduction To Affine Kac-moody Lie Algebras And Groups -- App. A. Results From Algebraic Geometry -- App. B. Local Cohomology. Shrawan Kumar. Includes Bibliographical References (p. [559]-589) And Indexes. Front Matter....Pages i-xv Kac-Moody Algebras....Pages 1-37 Representation Theory of Kac-Moody Algebras....Pages 39-65 Lie Algebra Homology and Cohomology....Pages 67-107 An Introduction to ind-Varieties and pro-Groups....Pages 109-148 Tits Systems....Pages 149-171 Kac-Moody Groups....Pages 173-197 Generalized Flag Varieties of Kac-Moody Groups....Pages 199-244 Demazure and Weyl-Kac Character Formulas....Pages 245-294 BGG and Kempf Resolutions....Pages 295-336 Defining Equations of G/P and Conjugacy Theorems....Pages 337-368 Topology of Kac-Moody Groups and Their Flag Varieties....Pages 369-445 Smoothness and Rational Smoothness of Schubert Varieties....Pages 447-479 An Introduction to Affine Kac Moody Lie Algebras and Groups....Pages 481-510 Back Matter....Pages 511-609