Induced conjugacy classes and induced U_e(G)-modules

TL;DR

This paper explores the relationship between conjugacy classes in a Lie group and the irreducible representations of a related quantum algebra, proposing a bijection and conjectures about their classification.

Contribution

It establishes a natural bijection between irreducible representations associated with conjugacy classes in the same Jordan class and proposes conjectures relating classes in the same sheet.

Findings

Bijection preserves dimension between representations in the same Jordan class

Conjectured relations for classes in the same sheet of G

Illustrated potential implications of these relations

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.