Verma modules over generalized Virasoro algebras Vir[G]

Let F be a field of characteristic 0, not necessarily algebraically closed, and G be an additive subgroup of F. For any total order on G which is compatible with the group addition, and for any , a Verma module over the generalized Virasoro algebra Vir[G] is defined. In the present paper, the irreducibility of Verma modules is completely determined.