Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups

Let $\text{Oq}(G)$ be the algebra of quantized functions on an algebraic group $G$ and $\text{Oq}(B)$ its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the “quantum flag variety of $G$” to be the $\text{Oq}(B)$-equivariant $\text{Oq}(G)$-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity.