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

2016/12/18 by Uchiyama, Tomohiro
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1612.05863

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 D₄ 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.

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