Branching laws on the metaplectic cover of ${\rm GL}_{2}$

Representation theory of $p$-adic groups naturally comes in the study of automorphic forms and one way to understand representations of a group is by restricting to its nice subgroups. D. Prasad studied the restriction for pairs $({\rm GL}_{2}(E), {\rm GL}_{2}(F))$ and $({\rm GL}_{2}(E), D_{F}^{\times})$ where $E/F$ is a quadratic equation and $D_{F}$ is the unique quaternion division algebra, and $D_{F}^{\times} \hookrightarrow {\rm GL}_{2}(E)$. Prasad proved a multiplicity one result and a `dichotomy' relating the restriction for the pairs $({\rm GL}_{2}(E), {\rm GL}_{2}(F))$ and $({\rm GL}_{2}(E), D_{F}^{\times})$ involving the Jacquet-Langlands correspondence. We study a restriction problem involving covering groups. In an analogy to the case of Prasad, we consider pairs $(\widetilde{{\rm GL}_{2}(E)}, {\rm GL}_{2}(F))$ and $(\widetilde{{\rm GL}_{2}(E)}, D_{F}^{\times})$ where $\widetilde{{\rm GL}_{2}(E)}$ is the $\mathbb{C}^{\times}$-metaplectic covering of ${\rm GL}_{2}(E)$. We do not have multiplicity one in this case but there is an analogue of dichotomy.