Local Weyl modules and skew Howe duality

Abstract

The skew $({\mathfrak{gl}}_n,{\mathfrak{gl}}_r)$ Howe duality states that the exterior algebra $\hat{I}›({\mathbb{C}}^{nr})$ admits a multiplicity-free decomposition under the natural actions of ${\mathfrak{gl}}_n\times {\mathfrak{gl}}_r$. In this paper, by using certain Lagrange interpolation polynomials of degree $r-1$, we extend the action of ${\mathfrak{gl}}_n$ on $Λ(\C^{nr})$ to its loop algebra $L({\mathfrak{gl}}_n)$. View $Λ(\C^{nr})$ as a module for the loop algebra $L({\mathfrak{sl}}_n)$ of ${\mathfrak{sl}}_n$ by taking restriction. We prove that every highest weight vector of ${\mathfrak{gl}}_n\times {\mathfrak{gl}}_r$ in $Λ(\C^{nr})$ generates a local Weyl module of $L({\mathfrak{sl}}_n)$. Furthermore, we obtain in this way an explicit realization of all local Weyl modules for $L({\mathfrak{sl}}_n)$.