Irreducible Virasoro modules from tensor products

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\mathrm{\Omega }(\lambda ,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta ,h)$ or with irreducible Virasoro modules Ind${}_{\theta }(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\mathrm{\Omega }(\lambda ,b)$ with a classical Whittaker module is isomorphic to the module ${\mathrm{I}\mathrm{n}\mathrm{d}}_{\theta ,\lambda }({\mathbb{C}}_{\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module ${\mathrm{I}\mathrm{n}\mathrm{d}}_{\theta ,\lambda }({\mathbb{C}}_{\mathbf{m}})$ to be irreducible. We also generalize the module $$\mathrm{I}\mathrm{n}\mathrm{d}${\theta, \lambda}(${\mathbb{C}}_{\mathbf{m}}$)$to$$\mathrm{I}\mathrm{n}\mathrm{d}${\theta,\lambda}($\mathcal{B}$^{(n)}_{$\mathbf{s}$})$ for any non-negative integer $n$ and use the above results to completely determine when the modules ${\mathrm{I}\mathrm{n}\mathrm{d}}_{\theta ,\lambda }({\mathcal{B}}_{\mathbf{s}}^{(n)})$ are irreducible. The submodules of ${\mathrm{I}\mathrm{n}\mathrm{d}}_{\theta ,\lambda }({\mathcal{B}}_{\mathbf{s}}^{(n)})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs’ Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.