On radical filtrations of parabolic Verma modules

In this paper we give a sum formula for the radical filtration of parabolic Verma modules in any (possibly singular) blocks of parabolic BGG category. It can be viewed as a generalization of the Jantzen sum formula for Verma modules in the usual BGG category O . The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of parabolic Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given in terms of the KazhdanLusztig polynomials.