Irreducible Virasoro modules from tensor products

Abstract

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\mathrm{\Omega }(\lambda ,b)$ with irreducible highest weight modules $V(\theta ,h)$ or with irreducible Virasoro modules ${\mathrm{Ind}}_{\theta }(N)$ defined in Mazorchuk and Zhao (Selecta Math. (N.S.) 20:839–854, 2014). 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 ${\mathrm{Ind}}_{\theta ,\lambda }({\mathbb{C}}_{\mathbf{m}})$ defined in Mazorchuk and Weisner (Proc. Amer. Math. Soc. 142:3695–3703, 2014). As a by-product we obtain the necessary and sufficient conditions for the module ${\mathrm{Ind}}_{\theta ,\lambda }({\mathbb{C}}_{\mathbf{m}})$ to be irreducible. We also generalize the module ${\mathrm{Ind}}_{\theta ,\lambda }({\mathbb{C}}_{\mathbf{m}})$ to ${\mathrm{Ind}}_{\theta ,\lambda }({\mathcal{B}}_{\mathbf{s}}^{(n)})$ for any non-negative integer $n$ and use the above results to completely determine when the modules ${\mathrm{Ind}}_{\theta ,\lambda }({\mathcal{B}}_{\mathbf{s}}^{(n)})$ are irreducible. The submodules of ${\mathrm{Ind}}_{\theta ,\lambda }({\mathcal{B}}_{\mathbf{s}}^{(n)})$ are studied and an open problem in Guo et al. (J. Algebra 387:68–86, 2013) is solved. Feigin–Fuchs’ Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.