Irreducible Virasoro modules from irreducible Weyl modules

We use Block’s results to classify irreducible modules over the differential operator algebra $\mathbb{C}[t,t^{-1},\frac{d}{dt}]$. From this classification and using “the twisting technique” we construct a lot of new irreducible modules over the Virasoro algebra. These new irreducible Virasoro modules are generally not weight modules. We determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided at the end of the paper. In particular the class of irreducible Virasoro modules $\mathrm{\Omega }(\lambda ,b)$ are defined on the polynomial algebra $\mathbb{C}[x]$.