Parabolic version of the two realizations theorem and applications to modular representation theory

The goal of this talk is two-fold. We state a parabolic version of the two realizations theorem and sketch a proof. This version relates Iwahori-constructible sheaves on parabolic affine flag variety to coherent sheaves on a parabolic version of the Steinberg. We also explain an application of this result to the representation theory of semisimple Lie algebras in positive characteristic. Our result here is an explicit character formula for appropriately equivariant simple modules with distinguished pp-character. The talk is based on 2005.10030, joint with Bezrukavnikov.