On Unipotent Radicals of Pseudo-Reductive Groups

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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.