Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 2

TL;DR

This paper investigates the concept of complete reducibility of subgroups in reductive algebraic groups over nonperfect fields, establishing new conditions and examples related to subgroup properties and conjugacy classes.

Contribution

It proves that the centralizer of a reductive k-subgroup is G-completely reducible over k if the subgroup is reductive, and shows that regular reductive k-subgroups are G-completely reducible over k.

Findings

Centralizer of a reductive k-subgroup is G-completely reducible if reductive.

Regular reductive k-subgroups are G-completely reducible.

Examples of infinite overgroups and conjugacy classes are provided.

Abstract

Let k be a separably closed field. Let G be a reductive algebraic k-group. In this paper, we study Serre's notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We also show that a regular reductive k-subgroup of G is G-completely reducible over k. Various open problems concerning complete reducibility are discussed. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.