Modular perverse sheaves on flag varieties I: tilting and parity sheaves

In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.