# Simple $\mathfrak{sl}(V)$-modules which are free over an abelian subalgebra

## TL;DR

This paper classifies certain simple modules over rak{sl}(V) that are free of rank 1 over a nilradical of a maximal parabolic subalgebra, parametrized by polynomials, and describes their structure and simplicity.

## Contribution

It provides a complete classification of rak{sl}(V)-modules that are free of rank 1 over a nilradical, including their parametrization and submodule structure.

## Key findings

- Modules are parametrized by polynomials in im V - 1 variables.
- Most of these modules are simple.
- Submodule structures are explicitly determined.

## Abstract

Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{sl}(V)$ of maximal dimension and let $\mathfrak{n} \subset \mathfrak{p}$ be the corresponding nilradical. In this paper we classify the set of $\mathfrak{sl}(V)$-modules whose restriction to $U(\mathfrak{n})$ is free of rank $1$. It turns out that isomorphism classes of such modules are parametrized by polynomials in $\dim V-1$ variables. We determine the submodule structure for these modules and we show that they generically are simple.