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 ${\Omega }_1$ 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 $W_n$.