Affine Schur Duality

The Schur duality may be viewed as the study of the commuting actions of the symmetric group S d and the general linear group GL(n, C) on E d where E = C n .Here we extend this duality to the context of the affine Weyl (or symmetric) group Z d S d and the affine Lie ( or Kac-Moody ) algebra g = Lg Cc, g = sl n (C).Thus we construct a functorS d ] -modules M to that of finite dimensional g -modules W of level 0 (the center Cc of g acts as zero, thus these are representations of the loop group Lg = L C g, where L = C[t, t -1 ], g = sl n (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 .