The solenoidal Heisenberg Virasoro algebra and its simple weight modules

Let Aₙ=C[tᵢ±1,~1≤ i≤ n] and W(n)_μ=Aₙd_μ the solenoidal Lie algebra introduced by Y.Billig and V.Futorny in {BiFu2}, where μ=(μ₁,…,μₙ)ⁿ is a generic vector and d_μ=∑ᵢ₌₁ⁿμᵢtᵢ∂/∂ tᵢ. We consider the semi-direct product Lie algebra WA(n)_μ:=W(n)_μ Aₙ. In the first part, We prove that WA(n)_μ has a unique three-dimensional universal central extension. In fact we construct a higher rank Heisenberg-Virasoro algebra (see {LiuGuo, LdZ}). It will be denoted by HVir(n)_μ and it will be called the solenoidal Heisenberg-Virasoro algebra. Then we will study Harish-Chandra modules of HVir(n)_μ following {LiuGuo}. We will obtain two classes of Harich-Chandra modules: generalized highest weight modules(GHW modules) and intermediate series modules. Our results are particular cases of {LiuGuo}. In the end, we will construct HVir(n)_μ Verma modules using the lexicographic order on Zⁿ. In particular we give examples of irreducible weight modules which have infinite dimensional weight spaces.