Affine schur duality

The Schur duality may be viewed as the study of the commuting actions of the symmetric group Sd and the general linear group GL(n,C) on E×d where E = Cn . Here we extend this duality to the context of the affine Weyl (or symmetric) group Zd oSd and the affine Lie ( or Kac-Moody) algebra eg = Lg⊕Cc, g = sln(C) . Thus we construct a functor F : M 7! M Sd E×d from the category of finite dimensional C[Zd o Sd] -modules M to that of finite dimensional eg -modules W of level 0 (the center Cc of eg acts as zero, thus these are representations of the loop group Lg = L C g, where L = C[t, t-1] , g = sln(C) ), the irreducible constituents of whose restriction to g are subrepresentations of E×d . When d < n it is an equivalence of categories, but not for d = n, in contrast to the classical case. As an application we conclude that all irreducible finite dimensional representations of Lg, the irreducible constituents of whose restriction to g are subquotients of E×d , are tensor products of evaluation representations at distinct points of C×.