Twisted $\mathcal{D}$-module extensions of local systems on a certain subvariety isomorphic to ${{\mathbb{G}}_{\text{m}}}^2$ of the affine flag variety of ${\text{SL}}_2$

Abstract

We introduce a family of rank-one local systems in the category of twisted $\mathcal{D}$-modules on a certain subvariety isomorphic to ${{\mathbb{G}}_{\text{m}}}^2$ of the affine flag variety of ${\text{SL}}_2$. We then give a criterion for these local systems, in terms of their parameters, to extend cleanly in the sense of $\mathcal{D}$-modules.