A Katsylo theorem for sheets of spherical conjugacy classes

Abstract

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.

Quotients for Sheets of Conjugacy Classes

“…Also in the group case one wants to reach a good understanding of quotients of sheets. An analogue of Katsylo's theorem was obtained for sheets containing spherical conjugacy classes and all such sheets are shown to be smooth [9]. The proof in this case relies on specific properties of the intersection of spherical conjugacy classes with Bruhat double cosets, which do not hold for general classes.…”

Affine hyperplane arrangements and Jordan classes

“…This stratification in Jordan classes for G has been crucial in the study of sheets for the adjoint action of G on itself [7]. An analogue of Katsylo's result [13] for S/G, where S is a sheet in G consisting of spherical conjugacy classes was given in [8]. In this case Slodowy slices are replaced by Sevostyanov's slices [19].…”