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

Abstract

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