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

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    $\begingroup$ 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. $\endgroup$ – Aaron Mar 19 '14 at 13:56
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    $\begingroup$ 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] $\endgroup$ – Peter Patzt Mar 20 '14 at 11:09
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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).

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