Integral affine Schur-Weyl reciprocity

Abstract

Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall algebra of the cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the modified quantum affine algebra of ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$. We will construct an integral form $\dot{{\mathfrak D}_{\vartriangle}}(n)$ for $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ such that the natural algebra homomorphism from $\dot{{\mathfrak D}_{\vartriangle}}(n)$ to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct the integral form ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$ of the universal enveloping algebra ${\mathcal U}(\hat{\frak{gl}}_n)$ of the loop algebra $\hat{\frak{gl}}_n=\frak{gl}_n({\mathbb Q})\otimes\mathbb Q[t,t^{-1}]$, and prove that the natural algebra homomorphism from ${\mathcal U}_\mathbb Z(\hat{\frak{gl}}_n)$ to the affine Schur algebra over $\mathbb Z$ is surjective.