We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This construction enables us to classify all simple Virasoro modules that are locally finite over a positive part. To obtain those irreducible Virasoro modules, we use simple modules over a family of finite dimensional solvable Lie algebras. For one of these algebras, all simple modules are classified by R. Block and we extend this classification to the next member of the family. As a result, we recover many known but also construct a lot of new simple Virasoro modules. We also propose a revision of the setup for study of Whittaker modules.

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Simple Virasoro modules which are locally finite over a positive part

Irreducible Virasoro modules from irreducible Weyl modules

U(h)-free modules and coherent families

Wn+- and Wn-module structures on U(hn)

A class of simple weight virasoro modules

Irreducible Virasoro modules from tensor products (II)

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ 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 $\Omega(\lambda,b)$ 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 $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ 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.

/pdf/irreducible-virasoro-modules-from-tensor-products-2r41fwu54u.pdf

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 $\Omega(\lambda,b)$ 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 $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ 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.

Haijun Tan

Papers

Wn+- and Wn-module structures on U(hn)

Irreducible Virasoro modules from tensor products (II)

Irreducible Virasoro modules from tensor products

Irreducible Virasoro modules from tensor products

Classification of simple locally finite modules over the affine-Virasoro algebras