Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 3

TL;DR

This paper investigates the rationality of Serre's complete reducibility for subgroups of reductive algebraic groups over nonperfect fields, providing new examples, theoretical insights, and geometric interpretations.

Contribution

It offers a theoretical foundation for previous constructions of subgroups with differing reducibility over nonperfect fields and introduces new examples and geometric invariant theory results.

Findings

Constructed new examples of subgroups with different reducibility properties over nonperfect fields.

Provided a theoretical framework explaining previous constructions.

Generalized results using Geometric Invariant Theory and topological methods.

Abstract

Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly non-connected) reductive group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In our previous work, we constructed examples of subgroups $H$ of $G$ that are $G$-completely reducible but not $G$-completely reducible over $k$ (and vice versa). In this paper, we give a theoretical underpinning of those constructions. To illustrate our result, we present a new such example in a non-connected reductive group of type $D_4$ in characteristic $2$. Then using Geometric Invariant Theory, we generalize the theoretical result above obtaining a new result on the structure of $G(k)$-(and $G$-) orbits in an arbitrary affine $G$-variety. We translate our result into the language of spherical buildings to give a topological viewpoint. A problem on centralizers of completely reducible subgroups and a problem concerning the number of conjugacy classes are also considered.