Category $\mathcal{O}$ for hybrid quantum groups and non-commutative Springer resolutions

The hybrid quantum group was firstly introduced by Gaitsgory, whose category $\mathcal{O}$ can be viewed as a quantum analogue of BGG category $\mathcal{O}$. We give a coherent model for its principal block at roots of unity, using the non-commutative Springer resolution defined by Bezrukavnikov--Mirkovi\'{c}. In particular, the principal block is derived equivalent to the affine Hecke category. As an application, we endow the principal block with a canonical grading, and show that the graded multiplicity of simple module in Verma module is given by the generic Kazhdan--Lusztig polynomial.