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

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 $\mathbb C^n$.