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

TL;DR

This paper investigates the restriction of representations from the metaplectic cover of GL(2) over a quadratic extension to subgroups, revealing an analogue of Prasad's dichotomy but without multiplicity one.

Contribution

It extends Prasad's restriction results to the setting of metaplectic covers of GL(2), establishing a dichotomy principle in this more complex context.

Findings

No multiplicity one in the metaplectic case

Existence of an analogue of Prasad's dichotomy

New insights into restriction problems for covering groups

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 $({\rm GL}_{2}(E), {\rm GL}_{2}(F))$ 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.