Mathematical Research Letters

Volume 30 (2023)

Number 5

On radical filtrations of parabolic Verma modules

Pages: 1485 – 1510

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n5.a7

Authors

Jun Hu (Key Laboratory of Algebraic Lie Theory and Analysis of Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China)

Wei Xiao (School of Mathematical Sciences, Shenzhen Key Laboratory of Advanced Machine Learning and Applications, Shenzhen University, Shenzhen, Guangdong, China)

Abstract

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 $\mathcal{O}$. 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 Kazhdan–Lusztig polynomials.

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The first author is supported by Natural Science Foundation of China (Grant No. 12171029). The second author is supported by the National Science Foundation of China (Grant No. 11701381 & 12371031) and Guangdong Natural Science Foundation (Grant No. 2017A030310138).

Received 21 June 2021

Received revised 29 March 2022

Accepted 17 May 2022

Published 14 May 2024