  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', Birkhauser 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.
