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

Abstract Let n ≥ 2 {n\geq 2} be an integer, 𝕊 n {\mathbb{S}_{n}} the Lie algebra of divergence zero vector fields on an n -dimensional torus, and 𝒦 n {\mathcal{K}_{n}} the Weyl algebra over the Laurent polynomial algebra A n = ℂ ⁢ [ x 1 ± 1 , x 2 ± 1 , … , x n ± 1 ] {A_{n}=\mathbb{C}[x_{1}^{\pm 1},x_{2}^{\pm 1},\dots,x_{n}^{\pm 1}]} . For any 𝔰 ⁢ 𝔩 n {\mathfrak{sl}_{n}} -module V and any module P over 𝒦 n {\mathcal{K}_{n}} , we define an 𝕊 n {\mathbb{S}_{n}} -module structure on the tensor product P ⊗ V {P\otimes V} . In this paper, necessary and sufficient conditions for the 𝕊 n {\mathbb{S}_{n}} -modules P ⊗ V {P\otimes V}