$lambda$-Differential operators and $lambda$-differential modules for the Virasoro algebra

The concept of $lambda$-differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the $lambda$-differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible $lambda$. Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the $lambda$-differential module structures on them.