A class of non-weight modules over the Virasoro algebra

For any triple $(\mu ,\lambda ,\alpha )$ of complex numbers and an $\mathfrak a$-module${V}$,aclassofnon-weightmodules$$\mathcal{M}$\big(V,\mu,\Omega(\lambda,\alpha)\big)$overtheVirasoroalgebra$\mathcal L$isconstructedinthispaper\mathrm{.}Weproveif$V$ 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 $\mathcal{M}(V,\mu ,\mathrm{\Omega }(\lambda ,\alpha ))$ is simple if and only if $\mu \mathrm{\ne }1,\lambda \mathrm{\ne }0,\alpha \mathrm{\ne }0,$. 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 $\mathcal{M}(V,\mu ,\mathrm{\Omega }(\lambda ,\alpha ))$ 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.