A family of simple weight modules over the Virasoro algebra

Using simple modules over the derivation Lie algebra $C[t]\frac{d}{d t}$ of the associative polynomial algebra $C[t]$, we construct new weight Virasoro modules with all weight spaces infinite dimensional. We determine necessary and sufficient conditions for these new weight Virasoro modules to be simple, and determine necessary and sufficient conditions for two such weight Virasoro modules to be isomorphic. If such a weight Virasoro module is not simple, we obtain all its submodules. In particular, we completely determine the simplicity and the isomorphism classes of the weight modules defined in [C. Conley, C. Martin; A family of irreducible representations of the Witt Lie algebra with infinite-dimensional weight spaces. Compos. Math., 128(2), 153-175(2001)] which are a small portion of the modules constructed in this paper.