On sheets of conjugacy classes in good characteristic

TL;DR

This paper classifies sheets of conjugacy classes in a reductive algebraic group over a field of good characteristic, establishing a bijection with certain triples involving centralizers, cosets, and unipotent classes.

Contribution

It provides a new classification of sheets in good characteristic using triples of algebraic data, extending known results to broader settings.

Findings

Sheets correspond to triples (M, Z(M)^ullet t, O)

Semisimple elements belong to unique sheets with O={1}

Unipotent classes in sheets relate to Levi subgroups

Abstract

We show that the sheets for a connected reductive algebraic group G over an algebraically closed field in good characteristic acting on itself by conjugation are in bijection with G-conjugacy classes of triples (M, Z(M)^\circ t, O) where M is the connected centralizer of a semisimple element in G, Z(M)^\circ t is a suitable coset in Z(M)/Z(M)^\circ and O is a rigid unipotent conjugacy class in M. Any semisimple element is contained in a unique sheet S and S corresponds to a triple with O={1}. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples for which M is a Levi subgroup of a parabolic subgroup of G and such a class is unique.