Abstract
Suppose that and are connected reductive groups over a number field and that an -homomorphism is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of to those of . If the adelic points of the algebraic groups , are replaced by their metaplectic covers, one may hope to specify an analogue of the -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 to those of . 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
Citation
Daniel Bump. Solomon Friedberg. David Ginzburg. "Lifting automorphic representations on the double covers of orthogonal groups." Duke Math. J. 131 (2) 363 - 396, 01 February 2006. https://doi.org/10.1215/S0012-7094-06-13126-5
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