Affine Schur Duality

Y. Z. Flicker

Journal of Lie theory, Tome 31 (2021) no. 3, pp. 681-718

Résumé

\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 $d$ 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 $\LL\fg$, the irreducible constituents of whose restriction to $\fg$ are subquotients of $\E^{\otimes d}$, are tensor products of evaluation representations at distinct points of $\C^\times$.

Détail

Classification : 17B10, 17B20, 17B65, 17B67, 22E50, 22E65, 22E67
Mots-clés : Affine Schur duality, affine Lie algebra, affine Kac-Moody algebra, loop group, loop algebra, affine Lie group, evaluation representations, finite dimensional representations