On the hyperalgebra of the loop algebra ${\widehat{\frak{gl}}_n}$

Abstract: Let $\widetilde{\mathcal U}{\mathbb Z}({\widehat{\frak{gl}}_n})$ be the Garland integral form of ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ introduced by Garland \cite{Ga}, where ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ is the universal enveloping algebra of ${\widehat{{\frak{gl}}}_n}$. Using Ringel--Hall algebras, a certain integral form ${\mathcal U}{\mathbb Z}(\widehat{{\frak{gl}}}n)$ of ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ was constructed in \cite{Fu13}. We prove that the Garland integral form $\widetilde{\mathcal U}{\mathbb Z}({\widehat{{\frak{gl}}}n})$ coincides with ${\mathcal U}{\mathbb Z}(\widehat{{\frak{gl}}}n)$. Let ${\mathpzc k}$ be a commutative ring with unity and let ${\mathcal U}{\mathpzc k}(\widehat{{\frak{gl}}}n)={\mathcal U}{\mathbb Z}(\widehat{{\frak{gl}}}n)\otimes{\mathpzc k}$. For $h\geq 1$, we use Ringel--Hall algebras to construct a certain subalgebra, denoted by ${{\mathtt{u}}}{!\vartriangle!}(n)h$, of ${\mathcal U}{\mathpzc k}(\widehat{{\frak{gl}}}n)$. The algebra ${{\mathtt{u}}}{!\vartriangle!}(n)h$ is the affine analogue of ${\mathtt{u}}({{\frak{gl}}}_n)_h$, where ${\mathtt{u}}({{\frak{gl}}}_n)_h$ is a certain subalgebra of the hyperalgebra associated with ${\frak{gl}}_n$ introduced by Humhpreys \cite{Hum}. The algebra ${\mathtt{u}}({{\frak{gl}}}_n)_h$ plays an important role in the modular representation theory of ${\frak{gl}}_n$. In this paper we give a realization of ${{\mathtt{u}}}{!\vartriangle!}(n)_h$ using affine Schur algebras.