Jantzen coefficients and radical filtrations for generalized Verma modules

TL;DR

This paper extends the Jantzen sum formula to generalized Verma modules in parabolic BGG categories, providing explicit radical filtration formulas and graded decomposition numbers, enhancing understanding of module structures.

Contribution

It introduces a sum formula for radical filtrations of generalized Verma modules and computes graded decomposition numbers in parabolic BGG categories.

Findings

Sum formula for radical filtrations of generalized Verma modules.

Explicit graded decomposition and inverse graded decomposition numbers.

Determination of radical filtrations using Jantzen coefficients.

Abstract

In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. 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 generalized Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given.