A class of simple weight Virasoro modules

For a simple module M   over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras arar) and any α∈Cα∈C, a class of weight modules N(M,α)N(M,α) over the Virasoro algebra is constructed. These weight modules have infinite dimensional weight spaces if dim⁡M>1dim⁡M>1. The necessary and sufficient conditions for N(M,α)N(M,α) to be simple are obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules Γ(α1,α2,λ1,λ2)Γ(α1,α2,λ1,λ2) are defined on the polynomial algebra C[x]⊗C[t,t−1]C[x]⊗C[t,t−1] for any α1,α2,λ1,λ2∈Cα1,α2,λ1,λ2∈C with λ1λ1 or λ2λ2 nonzero. By twisting the weight modules N(M,α)N(M,α) we also obtain nonweight simple Virasoro modules N(M,β)N(M,β) for any nonconstant β∈C[t,t−1]β∈C[t,t−1].