Reductive pairs arising from representations

Let G be a reductive algebraic group and V a G  -module. We consider the question of when (GL(V),ρ(G))(GL(V),ρ(G)) is a reductive pair of algebraic groups, where ρ is the representation afforded by V. We first make some observations about general G and V  , then specialise to the group SL2(K)SL2(K) with K algebraically closed of positive characteristic p. For this group we provide complete answers for the classes of simple and Weyl modules, the behaviour being determined by the base p   expansion of the highest weight of the module. We conclude by illustrating some of the results from the first section with examples for the group SL3(K)SL3(K).