Distinguished cusp forms are theta series
type: Item
instance of: scholarly article
title: Distinguished cusp forms are theta series
zbMATH Open document ID: 0689.10041
DOI: 10.1215/S0012-7094-89-05905-X
author: Jian-Shu Li
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The author studies theta liftings defined by the oscillator representation of the dual reductive pair \((Sp_{2n}, O_ n)\) over a number field k. The main result of the paper states that if \(\pi\) is an irreducible cuspidal automorphic representation of \(Sp_{2n}({\mathbb{A}}_ k)\) which is distinguished and associated to a fixed non-singular symmetric matrix \(T_ 0\) of size n with entries in k, then \(\pi\) is a constituent of the ``space of theta functions'' attached to \(T_ 0\). Here \(\pi\) is defined to be distinguished (and associated to \(T_ 0)\) if all functions f in the space of \(\pi\) have vanishing Fourier coefficients \(f_ T\) for all non-singular T which are not \(GL_ n(k)\)-equivalent to \(T_ 0\). Also the ``space of theta functions'' attached to \(T_ 0\), by definition, is the closed subspace of automorphic forms on \(Sp_{2n}({\mathbb{A}}_ k)\) generated by all theta-lifts defined by the oscillator representation of the dual pair \((Sp_{2n}, O_{T_ 0})\) where \(O_{T_ 0}\) is the orthogonal group attached to \(T_ 0.\) The author's result can be viewed as a generalization of the work of \textit{S. Gelbart} and \textit{I. I. Piatetski-Shapiro} [Invent. Math. 59, 145-188 (1980; Zbl 0426.10027)] who have characterized distinguished forms on \(\tilde Sp_ 2\) as theta series. The main ingredients of the proof are i) the classification of the distinguished representations of \(Sp_{2n}(F)\) where F is a local field, in terms of the unitary dual of \(O_{T_ 0}(F)\) and ii) the characterization of theta functions in terms of L-series due to Piatetski-Shapiro and Rallis.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 4128914
zbMATH Keywords: theta liftings
zbMATH Keywords: oscillator representation
zbMATH Keywords: dual reductive pair
zbMATH Keywords: cuspidal automorphic representation
zbMATH Keywords: space of theta functions
zbMATH Keywords: automorphic forms
zbMATH Keywords: L-series
reviewed by: Winfried Kohnen
MaRDI profile type: Publication
cites work: Q4200278
cites work: Distinguished representations and modular forms of half-integral weight
cites work: Q5552484
cites work: Q3912659
cites work: Q3856974
cites work: Asymptotic properties of unitary representations
cites work: Q3856976
cites work: Singular unitary representations of classical groups
cites work: A new way to get Euler products.
cites work: Langlands' Functoriality and the Weil Representation
cites work: Représentations supercuspidales du groupe metaplectique
cites work: Q4662289
cites work: The CAP representations of GSp (4,).
cites work: Shimura and Shintani correspondences
cites work: Sur certains groupes d'opérateurs unitaires
full work available at URL: s0012-7094-89-05905-x
OpenAlex ID: W2044476949
description: scientific article; zbMATH DE number 4128914