Simple modules over the Lie algebras of divergence zero vector fields on a torus

Let $n\ge 2$ be an integer, ${\mathcal{K}}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\mathbb{C}[x_1^{\pm 1},x_2^{\pm 1},\mathrm{.}\mathrm{.}\mathrm{.},x_n^{\pm 1}]$, and ${\mathbb{S}}_n$ the Lie algebra of divergence zero vector fields on an $n$-dimensional torus. For any ${\mathfrak{s}\mathfrak{l}}_n$-module $V$ and any module $P$ over ${\mathcal{K}}_n$, we define an ${\mathbb{S}}_n$-module structure on the tensor product $P\otimes V$. In this paper, necessary and sufficient conditions for the ${\mathbb{S}}_n$-modules $P\otimes V$ to be simple are given, and an isomorphism criterion for nonminuscule ${\mathbb{S}}_n$-modules is provided. More precisely, all nonminuscule ${\mathbb{S}}_n$-modules are simple, and pairwise nonisomorphic. For minuscule ${\mathbb{S}}_n$-modules, minimal and maximal submodules are concretely constructed.