  Let $G$ be a complex reductive group and $P$ be a parabolic subgroup of $G$.
In this paper the authors address questions involving the realization of the
$G$-module of the global sections of the (twisted) cotangent bundle over the
flag variety $G/P$ via the cohomology of the small quantum group. Our main
results generalize the important computation of the cohomology ring for the
small quantum group by Ginzburg and Kumar, and provides a generalization of
well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case
and the parabolic setting. As an application we answer the question (first
posed by Friedlander and Parshall for Frobenius kernels) about the realization
of coordinate rings of Richardson orbit closures for complex semisimple groups
via quantum group cohomology. Formulas will be provided which relate the
multiplicities of simple $G$-modules in the global sections with the dimensions
of extension groups over the large quantum group.
