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 Ω(λ,b)$\Omega(\lambda,b)$ with irreducible highest weight modules V(θ,h)$V(\theta ,h)$ or with irreducible Virasoro modules Indθ(N)$\mathrm{Ind}_{\theta}(N)$ 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 Ω(λ,b)$\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module Indθ,λ(Cm)$\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ 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 Indθ,λ(Cm)$\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ to be irreducible. We also generalize the module Indθ,λ(Cm)$\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ to Indθ,λ(Bs(n))$\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer n$n$ and use the above results to completely determine when the modules Indθ,λ(Bs(n))$\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of Indθ,λ(Bs(n))$\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ 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.