On the exponent of geometric unipotent radicals of pseudo-reductive groups

TL;DR

This paper establishes bounds on the exponent of the geometric unipotent radical of certain pseudo-reductive groups formed via Weil restriction, based on invariants of purely inseparable field extensions, with specific focus on general linear and simple groups.

Contribution

It provides new bounds for the exponent of the unipotent radical in pseudo-reductive groups constructed through Weil restriction, extending understanding in cases involving simple groups.

Findings

Bounds for the exponent of the unipotent radical are derived in terms of extension invariants.

Results are explicitly computed for the case G' = GL_n.

Applications to simple groups are discussed.

Abstract

Let $k'/k$ be a finite purely inseparable field extension and let $G'$ be a reductive $k'$-group. We denote by $G=\R_{k'/k}(G')$ the Weil restriction of $G'$ across $k'/k$, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical $\RR_{u}(G_{\bar{k}})$ in terms of invariants of the extension $k'/k$, starting with the case $G'=\GL_n$ and applying these results to the case where $G'$ is a simple group.