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

TL;DR

This paper investigates how often irreducible admissible representations of GL_2(F) appear within certain genuine representations of a non-trivial double cover of GL_2(E), where E/F is a quadratic extension, over a non-Archimedean local field.

Contribution

It provides new results on the multiplicity of representations in the context of metaplectic covers of GL_2 over quadratic extensions.

Findings

Determines multiplicity formulas for representations in the metaplectic setting.

Establishes conditions for the occurrence of irreducible representations.

Advances understanding of branching laws for covering groups.

Abstract

Let $F$ be a non-Archimedian local field of characteristic zero and $E/F$ a quadratic extension. The aim of the present article is to study the multiplicity of an irreducible admissible representation of ${\rm GL}_2(F)$ occurring in an irreducible admissible genuine representation of non-trivial two fold covering $\widetilde{{\rm GL}}_2(E)$ of ${\rm GL}_2(E)$.