Tensor modules over the Lie algebras of divergence zero vector fields on ${\mathbb{C}}^n$

Abstract

Let $n\ge 2$ be an integer, $S_n$ be the Lie algebra of vector fields on ${\mathbb{C}}^n$ with zero divergence, and $D_n$ be the Weyl algebra over the polynomial algebra $A_n=\mathbb{C}[t_1,t_2,\cdots ,t_n]$. In this paper, we study the simplicity of the tensor $S_n$-module $F(P,M)$, where $P$ is a simple $D_n$-module and $M$ is a simple ${\mathfrak{sl}}_n$-module. We obtain the necessary and sufficient conditions for $F(P,M)$ to be an irreducible module, and determine all simple subquotients of $F(P,M)$ when it is reducible.