Generalized Polynomial modules over the Virasoro algebra

Let $\mathcal{B}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible $\mathcal{B}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M, \Omega(\lambda,\beta))$ from irreducible $\mathcal{B}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $\Omega(\lambda,\beta)$. We give necessary and sufficient conditions for the Virasoro module $F(M, \Omega(\lambda,\beta))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M, \Omega(\lambda,\beta))$.