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, let $k'$ be a purely inseparable field extension of k of degree $p^e$ and let $G$ denote the Weil restriction of scalars $\mathrm{R}_{k'/k}(G')$ of a reductive $k'$-group $G'$. We prove that the unipotent radical $\mathscr{R}_u(G_{\bar k})$ of the extension of scalars of $G$ to the algebraic closure $\bar k$ of $k$ has exponent $e$. Our main theorem is to give bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases.