The description of regular blocks of the category O of a complex semisimple Lie algebra in terms of perverse sheaves on a flag variety has been a crucial tool for its study, and in particular for the proof of the Kazhdan-Lusztig character formula. This description has a conjectural analogue for representations of reductive groups over fields of positive characteristic, as predicted by Finkelberg-Mirkovic, which involves a category of perverse sheaves on the affine Grassmannian of the Langlands dual group, with coefficients in a field of positive characteristic. In this talk I will present a work in progress with Bezrukavnikov which we expect will lead to a proof of this conjecture. As a step towards this goal, we obtain a description of tilting perverse sheaves on the affine flag variety reminiscent of the corresponding result for usual flag varieties due to Soergel.