Special representations of rank one orthogonal groups
type: Item
instance of: scholarly article
title: Special representations of rank one orthogonal groups
zbMATH Open document ID: 0687.10020
DOI: 10.1007/BF02882424
author: I. I. Pyatetskii-Shapiro
author: David Soudry
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let G be the special orthogonal group of an m-dimensional quadratic space X with Witt index one over a number field k. The authors study (for \(m\geq 5)\) the image of the theta correspondence from \(\overline{Sl}_ 2({\mathbb{A}})\) to G(\({\mathbb{A}})\). The main result of the paper is a characterization of the image of this theta correspondence in terms of certain invariance properties of the Fourier coefficients of the cusp forms in this image. The cusp forms with this invariance property are called special. The proof of the fact that all special cusp forms occur in the image, requires first an analogous local statement (almost everywhere) and then the use of certain Rankin-Selberg integrals. A similar problem for the group PGSp(4)\(\simeq SO(3,2)\) was previously studied by the first-named author in [Prog. Math. 35, 309-325 (1983; Zbl 0566.10021)].
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 4125479
zbMATH Keywords: liftings
zbMATH Keywords: theta correspondence
zbMATH Keywords: invariance properties
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: cusp forms
zbMATH Keywords: special cusp forms
zbMATH Keywords: Rankin-Selberg integrals
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02882424
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description: scientific article; zbMATH DE number 4125479