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In this paper we give a sum formula for the radical filtration of generalizedVerma modules in any (possibly singular) blocks of parabolic BGG category whichcan be viewed as a generalization of Jantzen sum formula for Verma modules inthe usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, wedetermine the radical filtrations for all basic generalized Verma modules. Theproof makes use of the graded version of parabolic BGG category. Explicitformulae for the graded decomposition numbers and inverse graded decompositionnumbers of generalized Verma modules in any (possibly singular) integral blocksof the parabolic BGG category are also given.

Jantzen coefficients and radical filtrations for generalized Verma modules