A category equivalence on the Lie algebra of polynomial vector fields

Abstract

For any positive integer $n$, let $W_n=\text{Der}(\mathbb{C}[t_1,\dots,t_n])$. The subspaces $\mathfrak{h}_n=\text{Span}\{t_1\frac{\partial}{\partial{t_1}},\dots,t_n\frac{\partial}{\partial{t_n}}\}$ and $\Delta_n=\text{Span}\{\frac{\partial}{\partial{t_1}},\dots,\frac{\partial}{\partial{t_n}}\}$ are two abelian subalgebras of $W_n$. We show that a full subcategory $\Omega_{\mathbf{1}}$ of the category of $W_n$-modules $M$ which are locally finite over $\Delta_n$ is equivalent to some full subcategory of weight $W_n$-modules $M$ which are cuspidal modules when restricted to the subalgebra $\mathfrak{sl}_{n+1}$ of $W_n$.