Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

Abstract

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present the first example of a connected nonabelian $k$-subgroup $H$ of $G$ that is $G$-completely reducible but not $G$-completely reducible over $k$, and the first example of a connected nonabelian $k$-subgroup $H^{\prime }$ of $G$ that is $G$-completely reducible over $k$ but not $G$-completely reducible. This is new: all previously known such examples are for finite (or non-connected) $H$ and $H^{\prime }$ only.

arXiv admin note: text overlap with arXiv:1703.00103, arXiv:1612.05863. To appear in Journal of Group Theory