Lifting automorphic representations on the double covers of orthogonal groups

Abstract

Suppose that $G$ and $H$ are connected reductive groups over a number field $F$ and that an $L$-homomorphism $\rho :{^L G}{\longrightarrow}{^L H}$ is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of $G (\mathbb{A})$ to those of $H (\mathbb{A})$. 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 $\widetilde{{\rm SO}}_{2 k} (\mathbb{A})$ to those of $\widetilde{{\rm SO}}_{2 k + 1} (\mathbb{A})$. 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