Abstract

In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irreducible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group G. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class O are divisible by m^(1/2dimO). In this talk I shall outline a proof of an improved version of this conjecture and derive some important consequences of it related to q-W algebras.

A key ingredient of the proof are transversal slices S to the set of conjugacy classes in G. Namely, for every conjugacy class O in G one can find a special transversal slice S such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the real reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.