Restriction of representations of metaplectic GL₂(F) to tori

Let $F$ be a non-Archimedean local field. We study the restriction of an irreducible admissible genuine representations of the two fold metaplectic cover $\widetilde{GL}_{2}(F)$ of $GL_{2}(F)$ to the inverse image in $\widetilde{GL}_{2}(F)$ of a maximal torus in $GL_{2}(F)$.