Induced conjugacy classes and induced U_e(G)-modules

Abstract

By work of De Concini, Kac and Procesi the irreducible representations of the non-restricted specialization of the quantized enveloping algebra of the Lie algebra g at the roots of unity are parametrized by the conjugacy classes of a group G with Lie(G)=g. We show that there is a natural dimension preserving bijection between the sets of irreducible representations associated with conjugacy classes lying in the same Jordan class (decomposition class). We conjecture a relation for representations associated with classes lying in the same sheet of G, providing two alternative formulations. We underline some evidence and illustrate potential consequences.

This paper is an expanded version of a lecture given at the conference "Hopf algebras and tensor categories", Almeri'a, July 2011. It will appear in the volume of Contemporary Math. containing the conference Proceedings