Branching laws for the metaplectic cover of \(\operatorname{GL}_2\) (Q2406429)

Let \(F\) be a nonarchimedean local field of characteristic zero, and \(E/F\) be a quadratic extension. The author studies the multiplicity of an irreducible admissible representation of \(GL_2(F)\) occurring in an irreducible admissible genuine representation of the nontrivial two-fold covering \(\widetilde{\operatorname{GL}_2(E)}\) of \(\operatorname{GL}_2(E)\). The method extends the techniques of \textit{D. Prasad} [Am. J. Math. 114, No. 6, 1317--1363 (1992; Zbl 0780.22004)] who considered the representations of \(\operatorname{GL}_2(E)\).