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 $\Omega (\lambda ,b)$ 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 $\Omega (\lambda ,b)$ with a classical Whittaker module is isomorphic to the module ${\mathrm{Ind}}_{\theta ,\lambda }({\mathbb{C}}_{\mathbf{m}})$ defined in [MW]. 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 [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

arXiv comment17 Pages

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• Irreducible Virasoro modules from tensor products
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• $\lambda$-Differential operators and $\lambda$-differential modules for the Virasoro algebra
• Irreducible Virasoro modules from irreducible Weyl modules
• Irreducible modules over Witt algebras ${\mathcal{W}}_n$ and over ${\mathfrak{sl}}_{n+1}(\mathbb{C})$
• $\W_n^+$- and $W_n$-module structures on $U(h)$
• $U(\fh)$-free modules over the Block algebra $\BB(q)$
• New irreducible tensor product modules for the Virasoro algebra
• A class of non-weight modules over the Virasoro algebra
• Irreducible tensor product modules over the affine-Virasoro algebra of type $A_1$
• New irreducible non-weight Virasoro modules from tensor products