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

We study $\mathfrak{sl}(2)$-modules that are free of finite rank over $U(\mathfrak h)$, where $\mathfrak h$ is a fixed Cartan subalgebra of $\mathfrak{sl}(2)$.

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.