Multiplicity formula for restriction of representations of \(\widetilde{\mathrm{GL}_2}(F)\) to \(\widetilde{\mathrm{SL}_2}(F)\) (Q2789886)

Let \(F\) be a non-archimedean local field. Let \(\widetilde{\mathrm{SL}}_2(F)\) be the unique metaplectic cover of \(\mathrm{SL}_2(F)\). \(\mathrm{GL}_2(F)\) is the semi-direct product of \(\mathrm{SL}_2(F)\) and \(F^{\times}\), where \(F^{\times}\) embedded in \(\mathrm{GL}_2(F)\) as \(e \mapsto \begin{pmatrix} e & 0\\ 0 & 1 \end{pmatrix}\). This action of \(F^{\times}\) on \(\mathrm{SL}_2(F)\) lifts uniquely to an action of \(F^{\times}\) on \(\widetilde{\mathrm{SL}}_2(F)\). Denote \(\widetilde{\mathrm{GL}}_2(F) = \widetilde{\mathrm{SL}}_2(F) \rtimes F^{\times}\), called the metaplectic cover of \(\mathrm{GL}_2(F)\), which is one of the many inequivalent \(2\)-fold coverings of \(\mathrm{GL}_2(F)\) extending the \(2\)-fold covering of \(\mathrm{SL}_2(F)\).NEWLINENEWLINELet \(Z\) be the center of \(\mathrm{GL}_2(F)\), identified with \(F^{\times}\). Then \(\widetilde{Z}\) is the centralizer of \(\widetilde{\mathrm{SL}}_2(F)\) inside \(\widetilde{\mathrm{GL}}_2(F)\), but is not the center of \(\widetilde{\mathrm{GL}}_2(F)\). Let \(\widetilde{\mathrm{GL}}_2(F)_+ = \widetilde{Z} \cdot \widetilde{\mathrm{SL}}_2(F)\). Let \(\tau\) be an irreducible admissible genuine representation of \(\widetilde{\mathrm{SL}}_2(F)\) and \(\mu\) be a genuine character of \(\widetilde{Z}\), such that \(\mu |_{\widetilde{\{\pm 1\}}} = \omega_{\tau}\), the central character of \(\tau\). Then we can define a representation \(\mu \tau\) of \(\widetilde{\mathrm{GL}}_2(F)_+\) whose restriction to \(\widetilde{\mathrm{SL}}_2(F)\) is \(\tau\) and to \(\widetilde{Z}\) is \(\mu\). Consider NEWLINE\[NEWLINE\widetilde{\pi} := \mathrm{ind}_{\widetilde{\mathrm{GL}}_2(F)_+ }^{\widetilde{\mathrm{GL}}_2(F)} \mu \tau,NEWLINE\]NEWLINE which is an irreducible admissible genuine representation of \(\widetilde{\mathrm{GL}}_2(F)\). This paper studies the multiplicity of the restriction of \(\widetilde{\pi}\) to \(\widetilde{\mathrm{SL}}_2(F)\). Using results of Waldspurger on theta correspondence, the authors show that this multiplicity could be 2 or 4, in particular, may not be one, a result that was recently observed by Szpruch for certain principal series representations.