Irreducible Harish-Chandra modules over extended Witt algebras

Abstract

Let d be a positive integer, $A={\mathbb{C}} [t_{1}^{\pm1},\ldots ,t_{d}^{\pm1}]$ be the Laurent polynomial algebra, and $W=\operatorname{Der} (A)$ be the derivation Lie algebra of A. Then we have the semidirect product Lie algebra W⋉A which we call the extended Witt algebra of rank d. In this paper, we classify all irreducible Harish-Chandra modules over W⋉A with nontrivial action of A.