Automorphism groups of locally compact reductive groups

Abstract

A topological group G G is reductive if every continuous finite dimensional G G -module is semi-simple. We study the structure of those locally compact reductive groups which are the extension of their identity components by compact groups. We then study the automorphism groups of such groups in connection with the groups of inner automorphisms. Proposition. Let G G be a locally compact reductive group such that G / G 0 G/{G_0} is compact. Then I ( G 0 ) I\left ( {{G_0}} \right ) is dense in A 0 ( G ) {A_0}\left ( G \right ) .