Simple smooth modules over the Lie algebras of polynomial vector fields

Let $\mathfrak{g}:=\mathrm{D}\mathrm{e}\mathrm{r}(\mathbb{C}[t_1,t_2,\cdots ,t_n])$ and $\mathcal{L}:=\mathrm{D}\mathrm{e}\mathrm{r}(\mathbb{C}[[t_1,t_2,\cdots ,t_n]])$ be the Witt Lie algebras. Clearly, $\mathfrak{g}$ is a proper subalegbra of $\mathcal{L}$. Surprisingly, we prove that simple smooth modules over $\mathfrak{g}$ are exactly the simple modules over $\mathcal{L}$ studied by Rodakov (no need to take completion). Then we find an easy and elementary way to classify all simple smooth modules over $\mathfrak{g}$. When the height ${\mathrm{\ell }}_V\ge 2$ or $n=1$, any nontrivial simple smooth $\mathfrak{g}$-module $V$ is isomorphic to an induced module from a simple smooth ${\mathfrak{g}}_{\ge 0}$-module $V^{({\mathrm{\ell }}_V)}$. When ${\mathrm{\ell }}_V=1$ and $n\ge 2$, any such module $V$ is the unique simple quotient of the tensor module $F(P_0,M)$ for some simple ${\text{gl}}_n$-module $M$, where $P_0$ is a particular simple module over the Weyl algebra ${\mathcal{K}}_n^{+}$. We further show that a simple $\mathfrak{g}$-module $V$ is a smooth module if and only if the action of each of $n$ particular vectors in $\mathfrak{g}$ is locally finite on $V$.