Non-weight modules over the mirror Heisenberg-Virasoro algebra

Abstract

In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra \({\cal D}\), including Whittaker modules, \({\cal U}\left( {{\mathbb{C}d_0}} \right)\)-free modules and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all the \({\cal D}\)-module structures on \({\cal U}\left( {{\mathbb{C}d_0}} \right)\), and find the necessary and sufficient conditions for these modules to be irreducible. At last, we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and \({\cal U}\left( {{\mathbb{C}d_0}} \right)\)-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and \({\cal U}\left( {{\mathbb{C}d_0}} \right)\)-free modules are isomorphic. These lead to many new irreducible non-weight modules over \({\cal D}\).