Cubic correspondences arising from G2

We use the automorphic realization of the minimal representation (the theta representation) of the three fold cover of G₂ to construct a correspondence of automorphic forms between members of dual pairs inside G₂. The dual pairs under consideration are (SL₂, SL₂) and (SL₃,Z₃). In the first case, we obtain the Shimura correspondence between the three fold cover of SL₂ and SL₂, through explicit integral formulae. In the second case, we decompose theta over the three fold cover of SL₃, both locally and globally. The constituents are parametrized by their central characters and are cuspidal, except (essentially) when the central character is trivial. We thus obtain "cuspidal theta representations" and "the unramified theta representation" of the three fold cover of SL₃.