On the exponent of geometric unipotent radicals of pseudo-reductive groups

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Let k'/k be a finite purely inseparable field extension and let G' be a reductive k'-group. We denote by G=mathrm {R}_{k'/k}(G'), the Weil restriction of G' across k'/k, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical {mathscr {R}}_{u}(G_{bar{k}}) in terms of invariants of the extension k'/k, starting with the case G'={{,mathrm{GL},}}_n and applying these results to the case where G' is a simple group.