This volume is an attempt to provide an overview of some of the recent advances in representation theory from a geometric standpoint. A geometrically-oriented treatment is very timely and has long been desired, especially since the discovery of D-modules in the early '80s and the quiver approach to quantum groups in the early '90s.
The first half of the book fills the gap between the standard knowledge of a beginner in Lie theory and the much wider background needed by the working mathematician. Thus, Chapters 1-3 and 5-6 provide some basics in symplectic geometry, group actions on Kahler manifolds and Borel--Moore homology, geometry of semisimple groups, equivariant algebraic K-theory "from scratch," topology and algebraic geometry of flag varieties and conjugacy classes, respectively.
The material covered by Chapters 5 and 6 (as well as most of Chapter 3) has never been presented in book form. Chapters 3-4 and 7-8 form the heart of the book, presenting a uniform approach to representation theory of three quite different objects: (1) Weyl groups; (2) Lie algebra sln; (3) Iwahori--Hecke algebra. The results of Chapters 4 and 8 are new, with complete proofs, not to be found elsewhere in the literature. The techniques developed are quite general and can be successfully applied to such other areas of mathematics, as Quantum groups, affine Lie algebras, and quantum field theory. The exposition is practically self-contained and each chapter potentially serving as a basis for a graduate course.
