On supercuspidal representations of the metaplectic group

The Weil representations associated to anisotropic quadratic forms in one and three variables are used to study supercuspidal representations of the two-fold metaplectic covering group ${\overline {{\text {GL}}} _2}(k)$, where $k$ is a local nonarchimedean field of odd residual characteristic. The principal result is the explicit calculation of certain Whittaker functionals for any square-integrable irreducible admissible genuine representation of ${\overline {{\text {GL}}} _2}(k)$. In particular, a recent conjecture of Gelbart and Piatetski-Shapiro is answered by obtaining a bijection between the set of quasicharacters of ${k^ \ast }$ and the set of irreducible admissible genuine distinguished representations of ${\overline {{\text {GL}}} _2}(k)$, i.e. those representations possessing only one Whittaker functional, or, equivalently, those having a unique Whittaker model. The distinguished representations are precisely the representations attached to the Weil representation associated to a one dimensional form. The local piece of the generalized Shimura correspondence between automorphic forms of ${\overline {{\text {GL}}} _2}({\mathbf {A}})$ and ${\text {G}}{{\text {L}}_2}({\mathbf {A}})$ is also treated. Based upon a conjecture of the equivalences among the constituents of the Weil representations associated to two nonequivalent ternary forms, evidence for the explicit form of the local piece of this global correspondence, restricted to supercuspidal representations of ${\overline {{\text {GL}}} _2}(k)$, is presented. In this form, the map is shown to be injective and its image is described.