Verma modules for rank two Heisenberg-Virasoro algebra

Let ⪯ be a compatible total order on the additive group ℤ 2 , and L be the rank two Heisenberg-Virasoro algebra. For any c = ( c1, c2, c3, c4 ) ∈ ℂ 4 , we define a ℤ 2 -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.