Supercuspidal Duality for the Two-Fold Cover of SL 2 and the Split O 3

Abstract

Introduction. As first introduced the Shimura-Shintani-Niwa correspondence was a correspondence between homomorphic cusp forms of half-integral weight and certain modular forms of integral weight. Various authors have studied this correspondence in a representation-theoretic framework. Gelbart's seminal work [G1] provides some tools for translation and miscellaneous results. Using Whittaker models and L-functions, Gelbart and Piatetski-Shapiro [GPS1], [GPS2], obtained a generalized correspondence between automorphic cuspidal representations of GL2, the metaplectic group, and automorphic representations of GL2. Flicker [F] used a variation on the Selberg trace formula to extend the work of Gelbart and Piatetski-Shapiro. Finally, Waldspurger [W] returned to Whittaker models andL-functions to obtain a correspondence between automorphic cuspidal representations of SL2, the nontrivial two-fold cover of SL2, and automorphic representations of PGL2. Our more modest results complement the above and are obtained by purely algebraic means. We now attempt a brief summary.