Lifting free modules to generalized Weyl algebras

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules over a variety of algebras. In particular, we show that free modules that have rank $1$ over $R$ can be parametrized as $V_{\mathsf{p}}$ where $\mathsf{p}$ is a divisor of $a$. We give simplicity criteria for $V_{\mathsf{p}}$ and, additionally, when $R$ is a PID, provide a complete combinatorial description of the submodule structure of $V_{\mathsf{p}}$ and of the weight modules occurring as subquotients. We also show that, under some mild conditions on $R(\sigma,a)$, there exist simple $R$-free modules of arbitrary finite rank. We apply our results to $\mathfrak{sl}_2$ in order to construct new families of simple Cartan-free modules of all finite ranks.