A Katsylo theorem for sheets of spherical conjugacy classes

TL;DR

This paper establishes a geometric description of sheets of spherical conjugacy classes in reductive groups, showing their orbit spaces are quotients of affine varieties by finite abelian groups, and proves smoothness of these sheets in simple groups.

Contribution

It provides a new geometric characterization of sheets containing spherical conjugacy classes and proves their smoothness in simple algebraic groups.

Findings

Orbit space of a sheet is isomorphic to a quotient of an affine subvariety by a finite abelian group.

Sheets of spherical conjugacy classes in simple groups are always smooth.

The paper lists which strata containing spherical classes are smooth.

Abstract

We show that, for a sheet or a Lusztig stratum S containing spherical conjugacy classes in a connected reductive algebraic group G over an algebraically closed field in good characteristic, the orbit space S/G is isomorphic to the quotient of an affine subvariety of G modulo the action of a finite abelian 2-group. The affine subvariety is a closed subset of a Bruhat double coset and the abelian group is a finite subgroup of a maximal torus of G. We show that sheets of spherical conjugacy classes in a simple group are always smooth and we list which strata containing spherical classes are smooth.