Center of the category $\mathcal{O}$ for a hybrid quantum group

TL;DR

This paper establishes a deep connection between the center of category O for a hybrid quantum group at roots of unity and the cohomology of fixed loci on affine Grassmannians, with applications to Springer resolutions.

Contribution

It proves an algebra isomorphism linking the center of category O to affine Grassmannian cohomology and constructs an equivalence to sheaves on Springer resolutions for specific blocks.

Findings

Isomorphism between the center of category O and affine Grassmannian cohomology.

Construction of an abelian equivalence for the Steinberg block.

Extension of previous deformed isomorphism results.

Abstract

We establish an algebra isomorphism between the center of the category $\mathcal{O}$ for a hybrid quantum group at a root of unity $\zeta$ and the cohomology of $\zeta$-fixed locus on affine Grassmannian. A deformed version of this isomorphism was established in the previous paper of the author. For the Steinberg block of $\mathcal{O}$, we construct an abelian equivalence to the category of equivariant sheaves on the Springer resolution.