On Unipotent Radicals of Pseudo-Reductive Groups

Abstract

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive $k$-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let $k\text{'}$ be a purely inseparable field extension of $k$ of degree $p^e$, and let $G$ denote the Weil restriction of scalars ${\mathrm{R}}_{k\text{'}/k}(G\text{'})$ of a reductive $k\text{'}$-group $G\text{'}$. When $G={\mathrm{R}}_{k\text{'}/k}(G\text{'})$, we also provide some results on the orders of elements of the unipotent radical ${\mathcal{R}}_u(G_{\overline{k}})$ of the extension of scalars of $G$ to the algebraic closure $\overline{k}$ of $k$.