The subgroup structure of pseudo-reductive groups

TL;DR

This paper explores the subgroup structure of pseudo-reductive groups over fields, characterizing certain subgroups and their relation to reductive quotients, with implications for understanding maximal smooth subgroups.

Contribution

It provides new characterizations and conditions for the existence of subgroups in pseudo-reductive groups related to their reductive quotients.

Findings

Characterized smooth subgroups H with π'(H_{k'})=G'

Identified conditions for lifting subgroups from G' to G

Illustrated obstructions to subgroup existence with examples

Abstract

Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.