$\W_n^+$- and $W_n$-module structures on $U(h)$

Let $\h_n$ be the Cartan subalgebra of the Witt algebras $\W_n^+=\text{Der}\C[t_1, t_2, ..., t_n]$ and $\W_n=\text{Der}\C[t_1^{\pm 1},t_2^{\pm 1},\cdots,t_n^{\pm1}]$ where $1\le n\le \infty$. In this paper, we classify the modules over $\W_n^+$ and over $\W_n$ which are free $U(\h_n)$-modules of rank $1$. These are the $\W_n^+$-modules $Ω(Λ_{n},a, S)$ for some $Λ_n=(λ_1,\cdots,λ_n) \in (\C^*)^n, a\in \C$, and $S\subset \{1,2,..., n\}$; and $\W_n$-modules $Ω(Λ_n,a)$ for some $Λ_n\in (\C^*)^n$ and some $a\in \C.$

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