A parametrization of sheets of conjugacy classes in bad characteristic

TL;DR

This paper extends the parametrization of sheets of conjugacy classes in simple algebraic groups to bad characteristic, linking them to conjugacy classes of pairs involving centralizers and rigid unipotent classes.

Contribution

It introduces a new parametrization of sheets in bad characteristic, generalizing the known good characteristic case.

Findings

Sheets are parametrized by G-conjugacy classes of pairs (M,O).

M is the identity component of the centralizer of a semisimple element.

O is a rigid unipotent conjugacy class in M.

Abstract

Let G be a simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.