Combinatorics of Harish-Chandra modules

These lectures (given at a NATO-workshop 1997 in Montreal) survey recent work on the combinatorics of certain infinite dimensional representations of complex semisimple Lie algebras. Their focus is not on understanding the irreducible objects but rather on understanding the structure of suitable representation categories. They concentrate on the relation of these representation categories with categories of modules over the coinvariant algebra associated to the action of the Weyl group on a Cartan subalgebra. We also discuss conjectural generalizations to the representation theory of real Lie groups.