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

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 ${\rm PGL}_{2}(E)$.