Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1

TL;DR

This paper constructs examples of nonabelian subgroups of certain algebraic groups over nonperfect fields that are G-completely reducible over the algebraic closure but not over the base field, advancing understanding of subgroup reducibility in nonperfect settings.

Contribution

It provides the first known examples of non-G-completely reducible subgroups over nonperfect fields that become reducible over the algebraic closure, and establishes new results on complete reducibility over such fields.

Findings

Constructed explicit examples of non-G-completely reducible subgroups over nonperfect fields.

Proved that G-complete reducibility over the algebraic closure does not imply reducibility over the base field.

Established general results linking pseudo-reductivity and complete reducibility over nonperfect fields.

Abstract

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible $k$-subgroups of $G$ which are $G$-completely reducible over $k$. Our construction is based on that of subgroups of $G$ acting non-separably on the unipotent radical of a proper parabolic subgroup of $G$ in our previous work. We also present examples with the same property for a non-connected reductive group $G$. Along the way, several general results concerning complete reducibility over nonperfect fields are proved using the recently proved Tits center conjecture for spherical buildings. In particular, we show that under mild conditions a $k$-subgroup of $G$ is pseudo-reductive if it is $G$-completely reducible over $k$.