On conjugacy classes in a reductive group

TL;DR

This paper introduces a stratification of a connected reductive group into finitely many conjugacy class unions, and extends this to loop groups with a similar decomposition of regular semisimple elements, independent of the field's characteristic.

Contribution

It defines a new decomposition of reductive groups and loop groups into strata based on conjugacy class dimensions, independent of the characteristic.

Findings

Decomposition of G into finitely many strata indexed by Weyl group data.

Extension of the decomposition to loop groups for regular semisimple elements.

Strata are unions of conjugacy classes of fixed dimension.

Abstract

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set defined in terms of the Weyl group which is independent of the characteristic. In the case where $G$ is replaced by the corresponding loop group we define an analogous decomposition of the set of regular semisimple compact elements into countably many strata.