Mixed modular perverse sheaves on affine flag varieties and Koszul duality

Under some technical assumptions, and building on joint work with Bezrukavnikov, we prove a multiplicity formula for indecomposable tilting perverse sheaves on affine flag varieties, with coefficients in a field of characteristic $p$, in terms of $p$-Kazhdan–Lusztig polynomials. Under the same assumptions, we also explain the construction of a “degrading functor” relating mixed modular perverse sheaves (as defined in joint work with Achar) on such varieties to ordinary perverse sheaves.