Geometric methods in the representation theory of Hecke algebras and quantum groups

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the Iwahori-Hecke algebra of bi-invariant functions (under convolution) on a p-adic group, are considered. We give a uniform geometric construction of these algebras in terms of homology of an appropriate “Steinberg-type” variety Z (or its modification, such as K-theory or elliptic cohomology of Z, or an equivariant version thereof). We then explain how to obtain a complete classification of finite dimensional irreducible representations of the algebras in question, using our geometric construction and perverse sheaves methods. Similar techniques can be applied to other algebras, e.g. the double-affine Hecke algebras, elliptic algebras, quantum toroidal algebras.