On $U(\mathfrak{h})$-free modules of finite rank over $\mathfrak{s}\mathfrak{l}(2)$

We study $\mathfrak{s}\mathfrak{l}(2)$-modules that are free of finite rank over $U(\mathfrak{h})$, where $\mathfrak{h}$ is a fixed Cartan subalgebra of $\mathfrak{s}\mathfrak{l}(2)$. These modules form a natural class of non-weight modules. The coherent families obtained from this class via the weighting functor are identified. We also study a distinguished class of indecomposable $U(\mathfrak{h})$-free modules defined in terms of Jordan blocks and give a recursive description of their socle filtrations. Finally, we apply the general results to exponential modules arising from the first Weyl algebra and obtain simplicity criteria for these modules.