Reductive pairs arising from representations

We study the question of when a given rational representation of a reductive group
G gives rise to a reductive pair (GL(V ), ρ(G)), presenting complete classifications
when ρ is the representation afforded by a simple module for the group SL2 (K), or
a symmetric power of the natural module (the induced or dual Weyl modules for
this group), where K is an algebraically closed field of any positive characteristic.
We also present several classes of examples for the group SL3 (K) in some small
characteristics, along with results allowing new examples to be generated.