A Katsylo theorem for sheets of spherical conjugacy classes

Abstract

We show that, for a sheet or a Lusztig stratum S S containing spherical conjugacy classes in a connected reductive algebraic group G G over an algebraically closed field in good characteristic, the orbit space S / G S/G is isomorphic to the quotient of an affine subvariety of G G modulo the action of a finite abelian 2 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 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.