Representations of non-finitely graded Lie algebras related to Virasoro algebra

In this paper, we study representations of non-finitely graded Lie algebras $\mathcal{W}(\epsilon )$ related to Virasoro algebra, where $\epsilon =\pm 1$. Precisely speaking, we completely classify the free $\mathcal{U}(\mathfrak{h})$-modules of rank one over $\mathcal{W}(\epsilon )$,and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.