Quotients for sheets of conjugacy classes

TL;DR

This paper characterizes the orbit space of sheets of conjugacy classes in complex simple algebraic groups, establishing a bijection with a quotient of a shifted torus and analyzing the normality of categorical quotients.

Contribution

It provides a novel description of the orbit space of sheets for conjugation actions, extending previous results and including detailed examples like G2.

Findings

Describes the orbit space of sheets via a bijection with a quotient of a shifted torus.

Provides conditions for the normality of the categorical quotient of sheets.

Works out detailed example for the group G2.

Abstract

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S/G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient \overline{S}//G for arbitrary simple G and give a necessary and sufficient condition for S//G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G_2 is worked out in detail.