Affine Springer fiber and the small quantum group

TL;DR

The paper establishes a new geometric perspective on quantum group modules at roots of unity, linking them to microsheaves on affine Springer fibers and exploring related Langlands and mirror symmetry equivalences.

Contribution

It introduces a geometric realization of the principal block of quantum group modules as microsheaves on affine Springer fibers and connects it to Langlands and mirror symmetry frameworks.

Findings

Realization of the principal block as microsheaves on affine Springer fibers

Proved a geometric Langlands equivalence with wild ramification

Identified a homological mirror symmetry for the Springer resolution

Abstract

We find a new geometric incarnation for the principal block in the category of modules over a quantum group at a root of unity, realizing it as a full subcategory of microsheaves on a certain affine Springer fiber. We also prove a related geometric Langlands type equivalence with wild ramification, identifying the latter category with a category of coherent sheaves on the Springer resolution for the dual group. This can also be viewed as a version of homological mirror symmetry for the Springer resolution.