Schur-Weyl duality relates representations of the symmetric group to representations of $GL(n)$. Is there a generalization to arbitrary reductive groups?
As Aaron has pointed out, there are generalizations of the Schur-Weyl duality to certain reductive groups. But as far as I know, there is no unified theory. This paper of Stephen Doty seems to give a good overview of what can be done (although it is most probably not exhausting).