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 such that H is G-completely reducible but not G-completely reducible over k (and vice versa). This is new: all previously known such examples are for finite (or non-connected) H only.