Verma modules over the generalized Heisenberg-Virasoro algebra

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For any additive subgroup G of an arbitrary field F of characteristic zero, there corresponds a generalized Heisenberg-Virasoro algebra L[G]. Given a total order of G compatible with its group structure, and any h,hᵢ,c,cᵢ,cLI∈ F, a Verma module M(h,hᵢ,c,cᵢ,cLI) over L[G] is defined. In the this note, the irreducibility of Verma modules M(h,hᵢ,c,cᵢ,cLI) is completely determined.