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

Abstract: 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 $\Omega(\Lambda_{n},a, S) $ for some $\Lambda_n=(\lambda_1,\cdots,\lambda_n) \in (\C^*)^n, a\in \C$, and $S\subset {1,2,..., n}$; and $\W_n$-modules $\Omega(\Lambda_n,a)$ for some $\Lambda_n\in (\C^*)^n$ and some $a\in \C.$