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

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.