Lifting automorphic representations on the double covers of orthogonal groups

Suppose that G and H are connected reductive groups over a number field F and that an L-homomorphism ρ : ^L G → ^L H is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of G(double-struck A sign) to those of H(double-struck A sign). If the adelic points of the algebraic groups G, H are replaced by their metaplectic covers, one may hope to specify an analogue of the L-group (depending on the cover), and then one may hope to construct an analogous correspondence. In this article, we construct such a correspondence for the double cover of the split special orthogonal groups, raising the genuine automorphic representations of SO_2k(double-struck A sign) to those of SO_2k+1(double-struck A sign). To do so, we use as integral kernel the theta representation on odd orthogonal groups constructed by the authors in a previous article [3]. In contrast to the classical theta correspondence, this representation is not minimal in the sense of corresponding to a minimal coadjoint orbit, but it does enjoy a smallness property in the sense that most conjugacy classes of Fourier coefficients vanish.