Let $\mathfrak{ g}:={ \rm Der}(\mathbb{ C}[t_1, t_2,\cdots, t_n])$ and $\mathcal{ L}:={ \rm Der}(\mathbb{ C}[[t_1, t_2,\cdots, t_n]])$ 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 $\ell_{ V}\geq2$ or $n=1$, any nontrivial simple smooth $\mathfrak{ g}$-module $V$ is isomorphic to an induced module from a simple smooth $\mathfrak{ g}_{ \geq0}$-module $V^{ (\ell_{ V})}$. When $\ell_{ V}=1$ and $n\geq2$, any such module $V$ is the unique simple quotient of the tensor module $F(P_{ 0},M)$ for some simple $\gl_{ n}$-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 $V$.
Categories: math.RT math.QA math.RA