Whittaker category for the Lie algebra of polynomial vector fields

Abstract

For any positive integer $n$, let $A_n=\mathbb{C}[t_1,\dots ,t_n]$, $W_n=\text{Der}(A_n)$ and ${\Delta }_n=\text{Span}\{\frac{\partial }{\partial t_1},\dots ,\frac{\partial }{\partial t_n}\}$. Then $(W_n,{\Delta }_n)$ is a Whittaker pair. A $W_n$-module $M$ on which ${\Delta }_n$ operates locally finite is called a Whittaker module. We show that each block ${\Omega }_{\mathbf{a}}^{\tilde{W}}$ of the category of $(A_n,W_n)$-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over $L_n$, where $L_n$ is the Lie subalgebra of $W_n$ consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker $W_n$-modules with finite dimensional Whittaker vector spaces using ${\mathfrak{gl}}_n$-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block ${\Omega }_{\mathbf{a}}^W$.