On the conjugation action for quantum general linear groups

Abstract

We consider the conjugation action of a quantum group G over an arbitrary field k. In particular we consider the coordinate algebra k[G(n)] of the quantised general linear group G(n), at an arbitrary non-zero parameter q, as a G(n)-module and give analogues of results of Kostant and Richardson. We also consider the case in which G is a product of quantum general linear groups and the problem of describing the conjugation invariants of k[G] for the action of a quantum subgroup. This is approached via finite dimensional sub-coalgebras of k[G] and the theory of quasi-hereditary algebras.