$\mathfrak{so}_n(\mathbb{C})$-modules which are free over an abelian nilradical

Yang Chen; Haijun Tan; Lu Zhang

$\mathfrak{so}_n(\mathbb{C})$-modules which are free over an abelian nilradical

Representation Theory 2026-07-13 v1 Rings and Algebras

Authors: Yang Chen , Haijun Tan , Lu Zhang

Abstract

In this paper, we classify the category of $\mathfrak{so}_n(\mathbb{C})$ -modules whose restrictions to the universal enveloping algebra of the abelian nilradical of a maximal parabolic subalgebra are free of rank $1$ . We show that they generically are simple, and give an explicit description of submodules in the non-simple case. This category embodies a certain connection between weight modules and non-weight modules.

Cite

@article{arxiv.2607.11100,
  title  = {$\mathfrak{so}_n(\mathbb{C})$-modules which are free over an abelian nilradical},
  author = {Yang Chen and Haijun Tan and Lu Zhang},
  journal= {arXiv preprint arXiv:2607.11100},
  year   = {2026}
}