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

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.