Irreducible modules over the mirror Heisenberg-Virasoro algebra

In this paper, we study irreducible modules over the mirror Heisenberg-Virasoro algebra D, which is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. We classify all Harish-Chandra modules over D, i.e. irreducible modules with finite-dimensional weight spaces. Every such module is either an irreducible highest or an irreducible lowest weight module, or an irreducible module of the intermediate series. Furthermore, we use a twisted version of Feigin-Fuchs construction of the Virasoro algebra to establish the simplicity criterion for Verma modules and obtain a classification of unitary irreducible highest weight modules over D. Finally, we determine all irreducible restricted D-modules of level zero.