Complete Reducibility and Separability

TL;DR

This paper explores the relationship between separability and G-complete reducibility in reductive algebraic groups over fields of positive characteristic, demonstrating conditions under which subgroups are separable and their implications for conjugacy class intersections.

Contribution

It establishes that all subgroups are separable when the group is connected reductive and p is very good, and links G-complete reducibility to the semisimplicity of the Lie algebra as an H-module.

Findings

Subgroups are separable if G is connected reductive and p is very good.

G-complete reducibility is characterized by the semisimplicity of the Lie algebra as an H-module.

The analogue of Guralnick's conjugacy class result fails for n-tuples of elements when n > 1.

Abstract

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module. Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and H is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.