Classification of Simple Cuspidal Modules over Nongraded Witt Lie Algebras

Abstract

For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm 1},\dots ,t_n^{\pm 1},x_1,\dots ,x_n]$ and ${\mathfrak{g}}_n={⨁}_{i=1}^n{A}_n{d}_i$, where $d_i=t_i\frac{\partial }{\partial t_i}+\frac{\partial }{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple ${\mathfrak{gl}}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U({\mathfrak{g}}_n)\cong D_n\otimes U({\mathfrak{m}}_{1,0}\Delta )$, and use it to show that every simple cuspidal ${\mathfrak{g}}_n$-module is isomorphic to a simple quotient of some $T(A_n(\lambda ),V)$, where $V$ is a finite-dimensional simple ${\mathfrak{gl}}_n$-module.