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Representation theory of $p$-adic groups naturally comes in the study ofautomorphic forms and one way to understand representations of a group is byrestricting 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 quaterniondivision algebra, and $D_{F}^{\times} \hookrightarrow {\rm GL}_{2}(E)$. Prasadproved a multiplicity one result and a `dichotomy' relating the restriction forthe 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 tothe case of Prasad, we consider pairs $(\widetilde{{\rm GL}_{2}(E)}, {\rmGL}_{2}(F))$ and $(\widetilde{{\rm GL}_{2}(E)}, D_{F}^{\times})$ where$\widetilde{{\rm GL}_{2}(E)}$ is the $\mathbb{C}^{\times}$-metaplectic coveringof ${\rm GL}_{2}(E)$. We do not have multiplicity one in this case but there isan analogue of dichotomy.

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