A geometric realization of the center of the small quantum group

TL;DR

This paper introduces a geometric model for the center of the small quantum group using affine Springer fibers, establishing isomorphisms with cohomology and providing dimension bounds.

Contribution

It constructs a new geometric framework linking affine Springer fibers to the center of the small quantum group, including conjectured isomorphisms and dimension formulas.

Findings

Isomorphism between affine Springer fiber cohomology and the quantum group center.

Embedding of invariant cohomology into the quantum group center.

Dimension formula providing a lower bound for the center's size.

Abstract

We propose a new geometric model for the center of the small quantum group using the cohomology of certain affine Springer fibers. More precisely, we establish an isomorphism between the equivariant cohomology of affine Spaltenstein fibers for a split element and the center of the deformed graded modules for the small quantum group. We also obtain an embedding from the invariant part of the nonequivariant cohomology under the action of the extended affine Weyl group to the invariant part of the center of the small quantum group under Langlands dual group action, which we conjecture to be an isomorphism. Finally, we give a dimension formula for the invariants on the cohomology side, thus providing a lower bound for the dimension of the center.