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 pe and let G denote the Weil restriction of scalars Rk′/k(G′) of a reductive k′-group G′. We prove that the unipotent radical Ru(Gk¯) of the extension of scalars of G to the algebraic closure 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.