Pseudo-reductive and quasi-reductive groups over non-archimedean local fields

Abstract

Among connected linear algebraic groups, quasi-reductive groups generalize pseudo-reductive groups, which in turn form a useful relaxation of the notion of reductivity. We study quasi-reductive groups over non-archimedean local fields, focusing on aspects involving their locally compact topology. For such groups we construct valuated root data (in the sense of Bruhat--Tits) and we make them act nicely on affine buildings. We prove that they admit Iwasawa and Cartan decompositions, and we construct small compact open subgroups with an Iwahori decomposition. We also initiate the smooth representation theory of quasi-reductive groups. Among others, we show that their irreducible smooth representations are uniformly admissible, and that all these groups are of type I. Finally we discuss how much of these results remains valid if we omit the connectedness assumption on our linear algebraic groups.