Verma modules for rank two Heisenberg-Virasoro algebra

Let ⪯ be a compatible total order on the additive group ℤ2, and Lbe the rank two Heisenberg-Virasoro algebra. For any c= (c1, c2, c3, c4) ∈ℂ4, we define a ℤ2-graded Verma module M(c, ⪯) for L. A necessary and sufficient condition for M(c, ⪯) to be irreducible is provided. Moreover, the maximal ℤ2-graded submodules of M(c, ⪯) are characterized when M(c, ⪯) is reducible.