Multiplicity formula for restriction of representations of ${\widetilde{\mathrm{GL}}}_2(E)$ to ${\widetilde{\mathrm{SL}}}_2(E)$

Abstract

In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover ${\widetilde{\mathrm{GL}}}_2(E)$ of ${\mathrm{GL}}_2(E)$ to the 2-fold cover ${\widetilde{\mathrm{SL}}}_2(E)$ of ${\mathrm{SL}}_2(E)$, 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 ${\widetilde{\mathrm{SL}}}_2(E)$ and ${\mathrm{PGL}}_2(E)$.