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 $mathbb C^n$.