On the reducibility of scalar generalized Verma modules associated to two-step nilpotent parabolic subalgebras

2023/10/11 by Zhanqiang Bai, Bai, Zhanqiang, Fang, Minyan +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2310.07415

openalex publication_date 2023/10/11 · openalex created_date 2023/10/13 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakg be a simple complex Lie algebra.A generalized Verma module induced from a one-dimensional representation of a parabolic subalgebra of \mathfrakg is called a scalar generalized Verma module of \mathfrakg. In this article, we use Gelfand-Kirillov dimension to determine the reducibility of scalar generalized Verma modules of \mathfrakg associated to a two-step nilpotent parabolic subalgebra of non-maximal type. Such a module exists only when \mathfrakg=\mathfraksl(n,ℂ), \mathfrakso(2n,ℂ) or E₆. We find that the reducible points of these modules can be drawn in a two-dimensional complex plane.

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