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 be a purely inseparable field extension of k of degree pe, and let G denote the Weil restriction of scalars Rk/k(G) of a reductive k-group G. When G = Rk/k(G), we also provide some results on the orders of elements of the unipotent radical Ru(Gk¯ ) of the extension of scalars of G to the algebraic closure k¯ of k.