A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{s}\mathfrak{l}(2)$

We study simple $\mathfrak{s}\mathfrak{l}(2)$-modules over $\mathbb{C}$ that are free of finite rank as $U(\mathfrak{h})$-modules, where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{s}\mathfrak{l}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $G{L}_2(\mathbb{C}[h])$ and rely on Cohn’s standard form.