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

TL;DR

This paper investigates the representation theory of a specific hybrid quantum group at roots of unity, focusing on its category O, deformations, and the relationship between its center and equivariant cohomology of affine Grassmannian fixed points.

Contribution

It introduces the study of deformations of the category O for a hybrid quantum group and establishes an isomorphism between its center and equivariant cohomology of fixed loci.

Findings

Explicit computation of endomorphism algebra for big projective objects.

Construction of algebra isomorphism between the center of deformed category O and equivariant cohomology.

Analysis of subgeneric deformations of the category O.

Abstract

We study the representation theory of a hybrid quantum group at root of unity $\zeta$ introduced by Gaitsgory. After discussing some basic properties of its category $\mathcal{O}$, we study deformations of the category $\mathcal{O}$. For subgeneric deformations, we construct the endomorphism algebra of big projective object and compute it explicitly. Our main result is an algebra isomorphism between the center of deformed category $\mathcal{O}$ and the equivariant cohomology of $\zeta$-fixed locus on the affine Grassmannian attached to the Langlands dual group.