A class of non-weight modules over the Virasoro algebra

Abstract

For any triple $(\mu ,\lambda ,\alpha )$ of complex numbers and an $\mathfrak{a}$-module $V$, a class of non-weight modules $\mathcal{M}(V,\mu ,\Omega (\lambda ,\alpha ))$ over the Virasoro algebra $\mathcal{L}$ is constructed in this paper. We prove if $V$ is a nontrivial simple $\mathfrak{a}$-module satisfying: for any $v\in V$ there exists $r\in {\mathbb{Z}}_{+}$ such that $L_{r+i}v=0$ for all $i\ge 1$, then $\mathcal{M}(V,\mu ,\Omega (\lambda ,\alpha ))$ is simple if and only if $\mu \ne 1,\lambda \ne 0,\alpha \ne 0,$. We also give the necessary and sufficient conditions for two such simple $\mathcal{L}$-modules being isomorphic. Finally, we prove that these simple $\mathcal{L}$-modules $\mathcal{M}(V,\mu ,\Omega (\lambda ,\alpha ))$ are new by showing they are not isomorphic to any other known simple non-weight module provided that $V$ is not a highest weight $\mathfrak{a}$-module with highest weight nonzero.