Irreducible modules over the mirror Heisenberg–Virasoro algebra

Abstract

In this paper, we study irreducible modules over the mirror Heisenberg–Virasoro algebra [Formula: see text], which is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. We classify all Harish-Chandra modules over [Formula: see text], 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 [Formula: see text]. Finally, we determine all irreducible restricted [Formula: see text]-modules of level zero.