Component Groups of Unipotent Centralizers in Good Characteristic

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u ∈ G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checking, that the structure of the component group A(u) is independent of good characteristic. Throughout this note, Gwill denote a connected and reductive algebraic group G over the algebraically closed field k. For the most part, the characteristic p ≥ 0 of k is assumed to be good for G (see §1 for the definition). The main objective of our note is to extend the work of the second author [Som98] describing the component groups of unipotent (or nilpotent) centralizers. We recall a few definitions before stating the main result. A pseudo-Levi subgroup L of G is the connected centralizer C G(s) of a semisimple element s ∈ G. The reductive group L contains a maximal torus T of G, and so L is generated by T together with the 1 dimensional unipotent subgroups corresponding to a subsystem RL of the root system R of G; in §9 we make explicit which subsystems RL arise in this way when G is quasisimple. Let u ∈ G be a unipotent element, and let A(u) = CG(u)/C G(u) be the group of components (“component group”) of the centralizer of u. We are concerned with the structure of the group A(u) (more precisely: with its conjugacy classes). Consider the set of all triples