The Subgroup Structure of Pseudo-Reductive Groups

Abstract

Let k be a field. We investigate the relationship between subgroups of a pseudo-reductive k-group G and its maximal reductive quotient G′, with applications to the subgroup structure of G. Let k′/k be the minimal field of definition for the geometric unipotent radical of G, and let π′:Gk′→G′ be the quotient map. We first characterise those smooth subgroups H of G for which π′(Hk′)=G′. We next consider the following questions: given a subgroup H′ of G′, does there exist a subgroup H of G such that π′(Hk′)=H′, and if H′ is smooth can we find such a H that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup H, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of G with those of G′.