  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.
