In a pioneering 1981 paper De Concini and Procesi provided a beautiful description for cohomology of fixed point sets (Springer fibers) (G/B)_​ϵ in type A. This was an important precursor to two later developments: Khovanov's conjecture, proved by Brundan and Stroppel, identifying that cohomology with the center of a parabolic category O; and a conjectural generalization of the description of H∧∗((G/B)_​e) in the context of symplectic duality formulated by Khikita. I will mention a conceptual framework for Khovanov's conjecture and its generalizations beyond type A which involves a realization of Koszul dual to parabolic category O in terms of microlocal sheaves. The main results will have to do with a similar construction in the affine context, identifying Koszul dual to modules over the small quantum group with microlocal sheaves on an affine Springer fiber and describing the center of that category in terms of cohomology of an affine Springer fiber. Based on joint projects in progress with Boixeda Alvarez, McBreen and Yun and with Boixeda Alvarez, Shan and Vasserot.