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

Abstract Let 𝒲n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields. We classify simple weight 𝒲n {{\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 ℂn {\mathbb{C}^{n}} .