Simple Witt modules that are finitely generated over the Cartan subalgebra

Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the Weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, \dots, x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $V$ and any admissible module $P$ over the extended Witt algebra $\widetilde{W}_d$, we define a $W_d$-module structure on the tensor product $P\otimes V$. In this paper, we classify all simple $W_d$-modules that are finitely generated over the Cartan subalgebra. They are actually the $W_d$-modules $P \otimes V$ for a finite-dimensional simple $\mathfrak{gl}_d$-module $V$ and a simple $\mathcal{K}_d$-module $P$ that is a finite-rank free module over the polynomial algebra in the variables $x_1\frac{\partial}{\partial x_1},\dots,x_d\frac{\partial}{\partial x_d}$, except for a few cases which are also clearly described. We also characterize all simple $\mathcal{K}_d$-modules and all simple admissible $\widetilde{W}_d$-modules that are finitely generated over the Cartan subalgebra.