Let d ≥ 1 be an integer, W d and K d be the Witt algebra and the weyl algebra over the Laurent polynomial algebra, respectively. For any gl d -module M and any admissible module P over the extended Witt algebra W d , we define a W d -module structure on the tensor product P ⊗ M . We prove in this paper that any simple W d -module that is finitely generated over the cartan subalgebra is a quotient module of the W d -module P ⊗ M for a finite dimensional simple gl d -module M and a simple K d -module P that are finitely generated over the cartan subalgebra. We also characterize all simple K d -modules and all simple admissible W d -modules that are finitely generated over the cartan subalgebra.
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