Classification of irreducible Harish-Chandra modules over extended Divergence-zero Lie algebras

Let ${\mathcal{A}}_n=\text{C}[t_1^{\pm 1},t_2^{\pm 1},\dots ,t_n^{\pm 1}]$, and let $\text{EuScript}{D}_n$ denote the divergence-zero subalgebra of $\text{Der}({\mathcal{A}}_n)$. In this paper, we classify irreducible Harish-Chandra modules over the extended divergence-zero Lie algebra $\text{EuScript}G:=\text{EuScript}{D}_n⋉{\mathcal{A}}_n$ with nontrivial ${\mathcal{A}}_n^{\mathrm{\prime }}$-action, where ${\mathcal{A}}^{\mathrm{\prime }}n={\oplus }_{\mathbf{m}\in {\mathbb{Z}}^{\mathbf{n}}\setminus \{\mathbf{0}\}}\text{C}{t}^{\mathbf{m}}$. We prove that every such module is either cuspidal or a generalised highest weight module. We further prove that every irreducible generalised highest weight $\text{EuScript}G$-module is an irreducible highest weight module with respect to a suitable triangular decomposition of $\text{EuScript}G$. As a consequence, we obtain a classification of irreducible Harish-Chandra modules over $\text{EuScript}G$ with nontrivial ${\mathcal{A}}_n^{\mathrm{\prime }}$-action.