Generalized Schur-Weyl Duality

Schur-Weyl duality relates representations of the symmetric group to representations of $GL(n)$. Is there a generalization to arbitrary reductive groups?

• For $O(n)$ and $Sp(n)$, the algebra on the other side of the Schur-Weyl duality (replacing the group algebra $kS_n$) is called Brauer algebra, first studied in [H.Wenzl - On the structure of Brauer's centralizer algebras]; Goodman-Wallach's book also has an account on this subject. I don't know about general theory for reductive group though. – Aaron Mar 19 '14 at 13:56
• I guess as Wenzl points out in his paper, Brauer first studied those centralizer algebras in [R. BRAUER, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 854-872] – Peter Patzt Mar 20 '14 at 11:09