Component groups of unipotent centralizers in good characteristic

Abstract

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in 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.Comment: 13 pages; AMS LaTeX. This is the final version; it will appear in the Steinberg birthday volume of the Journal of Algebra. This version corrects an oversight pointed out by the referee; see Prop 2