Duke Mathematical Journal

Lifting automorphic representations on the double covers of orthogonal groups

Daniel Bump, Solomon Friedberg, and David Ginzburg

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Abstract

Suppose that G and H are connected reductive groups over a number field F and that an L-homomorphism ρ:LGLH is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of G(A) to those of H(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 SO~2k(A) to those of SO~2k+1(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

Article information

Source
Duke Math. J., Volume 131, Number 2 (2006), 363-396.

Dates
First available in Project Euclid: 12 January 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1137077888

Digital Object Identifier
doi:10.1215/S0012-7094-06-13126-5

Mathematical Reviews number (MathSciNet)
MR2219245

Zentralblatt MATH identifier
1107.11024

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F27: Theta series; Weil representation; theta correspondences 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings [See also 20G05] 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]

Citation

Bump, Daniel; Friedberg, Solomon; Ginzburg, David. Lifting automorphic representations on the double covers of orthogonal groups. Duke Math. J. 131 (2006), no. 2, 363--396. doi:10.1215/S0012-7094-06-13126-5. https://projecteuclid.org/euclid.dmj/1137077888


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