## Abstract

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×.

Original language | English |
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Pages (from-to) | 681-718 |

Number of pages | 38 |

Journal | Journal of Lie Theory |

Volume | 31 |

Issue number | 3 |

State | Published - 2021 |

## Keywords

- Affine Kac-Moody algebra
- Affine Lie algebra
- Affine Lie group
- Affine Schur duality
- Evaluation representations
- Finite dimensional representations
- Loop algebra
- Loop group