A Bessel identity for the theta correspondence
type: Item
instance of: scholarly article
title: A Bessel identity for the theta correspondence
zbMATH Open document ID: 1334.11031
author: Ehud Moshe Baruch
author: Erez Lapid
author: Zhengyu Mao
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: In this paper, the authors prove certain global identity between Bessel distributions with respect to generic cuspidal representations of an odd orthogonal group and the metaplectic cover of a symplectic group (of the same semisimple rank as this orthogonal group), which are related through the theta correspondence. They also prove the local counterpart of this result (and actually this calculation comprises the bulk of the paper), in the case of square integrable generic representations of the groups in question. Using some previous results of (some of the) authors, and other people (older results of Casselman, etc.) -- mainly about estimates of the values of Whittaker functions on split maximal torus -- they explicitly realize a generic, square-integrable representation in the space of Whittaker functions. They prove that this realization is actually in the space of square-integrable Whittaker functions (cf. section 2). Using these realizations, they explicitly realize generic theta correspondence in the equal rank-case -- as the bijection given by an integral transform -- formally very similar to the global theta integral (in the case of square-integrable representations). Then, using these results, and introducing Bessel distributions, they also prove the identity of Bessel distributions for matching test-functions (i.e., with compatible orbital integrals). In the last part of the paper, the authors prove analogous global results; first they prove that theta kernels give bijection between those generic cuspidal automorphic representations (on both global groups) for which the theta lift is non-zero. They also characterize which are the representations which have the non-zero theta lifts. Also this prepares the final global result, which is an appropriate analog of the matching in the local case.
reviewed by: Marcela Hanzer
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 6176544
zbMATH Keywords: theta correspondence
zbMATH Keywords: Bessel distributions
zbMATH Keywords: generic representations
MaRDI profile type: Publication
full work available at URL: s11856-012-0097-x
OpenAlex ID: W2089391963
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DOI: 10.1007/S11856-012-0097-X
description: scientific article; zbMATH DE number 6176544