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

TL;DR

This paper provides the first known examples of connected nonabelian subgroups of a reductive algebraic group over a nonperfect field that exhibit different notions of complete reducibility over the field and algebraic closure, highlighting new rationality phenomena.

Contribution

It introduces novel examples of connected nonabelian subgroups demonstrating the failure of rational complete reducibility over nonperfect fields, expanding understanding of subgroup behavior in algebraic groups.

Findings

First example of a connected nonabelian subgroup that is G-completely reducible but not over k.

First example of a connected nonabelian subgroup G-completely reducible over k but not over its algebraic closure.

Highlights new rationality phenomena in subgroup reducibility over nonperfect fields.

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 $H'$ 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 $H'$ only.