Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

TL;DR

This paper classifies simple weight modules with finite multiplicities over the Lie algebra of polynomial vector fields, showing they are either tensor modules or related to the de Rham complex.

Contribution

It provides a complete classification of simple weight modules with finite multiplicities over ${ m W}_n$, identifying their structure as tensor modules or de Rham complex modules.

Findings

All such modules are either tensor modules or submodules of tensor modules linked to the de Rham complex.

The classification is exhaustive for simple weight modules with finite multiplicities over ${ m W}_n$.

Abstract

Let ${\mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${\mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $\mathbb C^n$.