Multiplicity formula for restriction of representations of $\widetilde{{\rm GL}}_{2}(E)$ to $\widetilde{{\rm SL}}_{2}(E)$

TL;DR

This paper establishes a multiplicity formula for the restriction of irreducible genuine representations from a 2-fold cover of GL(2) to a 2-fold cover of SL(2), revealing that the multiplicity can exceed one, using Waldspurger's theta correspondence analysis.

Contribution

It provides the first explicit multiplicity formula for these restrictions, showing that the multiplicity may be greater than one, which was previously unnoticed.

Findings

Multiplicity can be greater than one.

Uses Waldspurger's theta correspondence analysis.

Provides explicit restriction multiplicity formula.

Abstract

In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover $\widetilde{{\rm GL}}_{2}(E)$ of ${\rm GL}_{2}(E)$ to the 2-fold cover $\widetilde{{\rm SL}}_{2}(E)$ of ${\rm SL}_{2}(E)$, and find in particular that this multiplicity may not be one, a result that seems to have been noticed before. The proofs follow the standard path via Waldspurger's analysis of theta correspondence between $\widetilde{{\rm SL}}_{2}(E)$ and ${\rm PGL}_{2}(E)$.