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

Abstract

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 $({\mathrm{GL}}_2(E),{\mathrm{GL}}_2(F))$ and $({\mathrm{GL}}_2(E),D_F^{\times })$ where $E/F$ is a quadratic equation and $D_F$ is the unique quaternion division algebra, and $D_F^{\times }\hookrightarrow {\mathrm{GL}}_2(E)$. Prasad proved a multiplicity one result and a `dichotomy' relating the restriction for the pairs $({\mathrm{GL}}_2(E),{\mathrm{GL}}_2(F))$ and $({\mathrm{GL}}_2(E),D_F^{\times })$ involving the Jacquet-Langlands correspondence. We study a restriction problem involving covering groups. In an analogy to the case of Prasad, we consider pairs $(\widetilde{{\mathrm{GL}}_2(E)},{\mathrm{GL}}_2(F))$ and $(\widetilde{{\mathrm{GL}}_2(E)},D_F^{\times })$ where $\widetilde{{\mathrm{GL}}_2(E)}$ is the ${\mathbb{C}}^{\times }$-metaplectic covering of ${\mathrm{GL}}_2(E)$. We do not have multiplicity one in this case but there is an analogue of dichotomy.

Ph. D. Thesis