Modules of Differential Operators on the Real Line

The space $\mathcal{D}^k$ of $k$th-order linear differential operators on $\mathbb{R}$ is equipped with a natural two-parameter family of structures of $\operatorname{Diff}(\mathbb{R})$-modules. To specify this family, one considers the action of differential operators on tensor densities. We give a classification of these modules.