Branching laws on the metaplectic cover of ${\mathrm{G}\mathrm{L}}_2$

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 $({\mathrm{G}\mathrm{L}}_2(E),{\mathrm{G}\mathrm{L}}_2(F))$ and $({\mathrm{G}\mathrm{L}}_2(E),D_F^{\times })$ where $E\mathrm{/}F$ is a quadratic equation and $D_F$ is the unique quaternion division algebra, and $D_F^{\times }\hookrightarrow {\mathrm{G}\mathrm{L}}_2(E)$. Prasad proved a multiplicity one result and a `dichotomy’ relating the restriction for the pairs $({\mathrm{G}\mathrm{L}}_2(E),{\mathrm{G}\mathrm{L}}_2(F))$ 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$\widetilde{{\rm GL}_{2}(E)}$isthe$$\mathbb{C}$^{\times}$-metaplectic covering of ${\mathrm{G}\mathrm{L}}_2(E)$. We do not have multiplicity one in this case but there is an analogue of dichotomy.