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Journal of Lie Theory 31 (2021), No. 3, 681--718
Copyright Heldermann Verlag 2021

Affine Schur Duality

Yuval Z. Flicker
Ariel University, Ariel 40700, Israel
and: The Ohio State University, Columbus, OH 43210-1174, U.S.A.


\def\GL{\operatorname{GL}} \def\C{\mathbb C} \def\E{\mathbb E} \def\Z{\mathbb Z} \def\fg{\mathfrak{g}} \def\LL{\mathcal L} \def\wt{\widetilde} \def\sll{\operatorname{sl}} \def\FF{\mathcal F} 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 $\E^{\otimes d}$ where $\E=\C^n$. Here we extend this duality to the context of the affine Weyl (or symmetric) group $\Z^d\rtimes S_d$ and the affine Lie $($or Kac-Moody$)$ algebra $\wt{\fg}=\LL\fg\oplus\C c$, $\fg=\sll_n(\C)$. Thus we construct a functor $\FF: M\mapsto M\otimes_{S_d}\E^{\otimes d}$ from the category of finite dimensional $\C[\Z^d\rtimes S_d]$-modules $M$ to that of finite dimensional\break $\wt{\fg}$-modules $W$ of level 0 (the center $\C c$ of $\wt{\fg}$ acts as zero, thus these are representations of the loop group $\LL\fg=\LL\otimes_{\C}\fg$, where $\LL=\C[t,t^{-1}]$, $\fg=\sll_n(\C)$), the irreducible constituents of whose restriction to $\fg$ are subrepresentations of $\E^{\otimes d}$. When $d
Keywords: Affine Schur duality, affine Lie algebra, affine Kac-Moody algebra, loop group, loop algebra, affine Lie group, evaluation representations, finite dimensional representations.

MSC: 17B10, 17B20, 17B65, 17B67, 22E50, 22E65, 22E67.

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