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 $\text{SL}_2$

TL;DR

This paper constructs and analyzes a family of rank-one local systems within twisted $ abla$-modules on a specific subvariety of the affine flag variety of SL₂, providing criteria for their extension.

Contribution

It introduces a new family of local systems on a subvariety of the affine flag variety and establishes criteria for their clean extension as twisted $ abla$-modules.

Findings

Defined a family of rank-one local systems on the subvariety.

Provided a criterion for the extension of these local systems.

Characterized parameters for clean extension in the $ abla$-module category.

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.