Verma modules over Virasoro-like algebras

Let K be a field of characteristic 0, G the direct sum of two copies of the additive group of integers. For a total order < on G, which is compatible with the addition, and for any (c) over dot(1), (c) over dot(2) is an element of K, we define G-graded highest weight modules M((c) over dot(1), (c) over dot(2), <) over the Virasoro-like algebra (L) over tilde, indexed by G. It is natural to call them Verma modules. In the present paper, the irreducibility of M((c) over dot(1), (c) over dot(2), <) is completely determined and the structure of reducible module M((c) over dot(1), (c) over dot(2), <) is also described.