Simple $\mathfrak{sl}_2$-modules that are torsion free $U(\mathfrak{h})$-modules of rank $1$

Abstract

We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.