A conjecture on Whittaker-Fourier coefficients of cusp forms
type: Item
instance of: scholarly article
title: A conjecture on Whittaker-Fourier coefficients of cusp forms
zbMATH Open document ID: 1396.11081
author: Zhengyu Mao
author: Erez Lapid
published in: Journal of Number Theory
publication date: 2014-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.3190
review text: Let \(G\) be a quasi-split reductive group over a number field \(F\), \(B\) be a Borel subgroup defined over \(F\), \(N\) its unipotent radical, and \(\psi_N\) a non-degenerate character of \(N(F)\backslash N(\mathbb{A})\). If \(\pi\) is a \(\psi_N\)-generic cuspidal automorphic representation of \(G(\mathbb{A})\), then the authors introduce a quantity \(c_\pi^{\psi_N}\) which measures the size of the \(\psi_N\)-Whittaker-Fourier coefficient of a matrix coefficient of \(\pi\) divided by a suitable product of Whittaker coefficients for the corresponding forms in the spaces \(\pi\) and \(\pi^\vee\). The precise definition requires some regularizations, which are treated in the paper. Using Rankin-Selberg theory, the authors show that this quantity is \(1\) for irreducible cuspidal representations of \(GL_n\); a similar result was proved independently by \textit{Y. Sakellaridis} and \textit{A. Venkatesh} [Periods and harmonic analysis on spherical varieties. Paris: Société Mathématique de France (SMF) (2017; Zbl 1479.22016)]. They then provide a conjectured value for the quantity more generally, based on Arthur's conjectures for the discrete spectrum, in the case of elliptic Arthur parameters of Ramanujan type. They reduce their conjecture, on some natural compatibility assumptions on Arthur's conjecture, to the case that \(G\) is semisimple and simply connected. They also formulate the conjecture more concretely when \(G\) is quasi-split and classical, by tying it to descent, and formulate a version for the metaplectic double cover of the symplectic group. The conjectures are also proved in some low rank cases. The conjectures of this paper may be viewed as an analogue for Whittaker coefficients of the conjectures of \textit{A. Ichino} and \textit{T. Ikeda} [Geom. Funct. Anal. 19, No. 5, 1378--1425 (2010; Zbl 1216.11057)], which concern periods of automorphic forms over closed subgroups and which sharpen the global Gross-Prasad conjecture. They also strengthen the con…
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 6359743
zbMATH Keywords: Whittaker coefficient
zbMATH Keywords: matrix coefficient
zbMATH Keywords: generic automorphic form
zbMATH Keywords: Arthur parameter
zbMATH Keywords: descent
MaRDI profile type: Publication
OpenAlex ID: W2060517985
Wikidata QID: Q122917410
arXiv ID: 1309.3190
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