Metaplectic theta functions and global integrals

We convolve a theta function on an n  -fold cover of GL3GL3 with an automorphic form on an n′n′-fold cover of GL2GL2 for suitable n  , n′n′. To do so, we induce the theta function to the n  -fold cover of GL4GL4 and use a Shalika integral. We show that when n=n′=3n=n′=3 this construction gives a new Eulerian integral for an automorphic form on the 3-fold cover of GL2GL2, and when n=4n=4, n′=2n′=2, it gives a Dirichlet series with analytic continuation and functional equation that involves both the Fourier coefficients of an automorphic form of half-integral weight and quartic Gauss sums. The analysis of these cases is based on the uniqueness of the Whittaker model for the local exceptional representation. The constructions studied here may be put in the context of a larger family of global integrals constructed using automorphic representations on covering groups. We sketch this wider context and some related conjectures.