On Sheets of Conjugacy Classes in Good Characteristic

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-orbits of triples $(M,\ Z(M)^\circ t,\ \mathcal O)$ where M is the connected centralizer of a semisimple element in G, Z(M)^°t is a suitable coset in Z(M)/Z(M)^° and $\mathcal O$ is a rigid unipotent conjugacy class in M; or, equivalently they are in bijection with G-orbits of pairs $(L,\ \mathcal O)$ with L a Levi factor of a parabolic subgroup of G and $\mathcal O$ a rigid conjugacy class of [L,L]. Any semisimple element is contained in a unique sheet S, and S corresponds to a triple with $\mathcal O=\{1\}$. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples for which M is a Levi factor of a parabolic subgroup of G and the class is unique.