New simple modules for the Heisenberg-Virasoro algebra

We construct a large class of simple modules for the Heisenberg-Virasoro algebra. These modules are determined by the simple modules over the finite-dimensional quotient algebras of certain natural subalgebras. The known simple modules such as the highest weight modules and Whittaker modules are included. In particular, we recover one of the main results of Billig (2003) in [3] and reobtain the simplicity of the Verma modules in a special case of Arbarello et al. (1988) in [1]. We also characterize a large subclass of these modules as simple modules with locally finite actions of elements in certain positive part.