Metaplectic forms
type: Item
instance of: scholarly article
title: Metaplectic forms
zbMATH Open document ID: 0559.10026
DOI: 10.1007/BF02698770
published in: Publications Mathématiques
publication date: 1984-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/103999
review text: The authors prove a Siegel-Weil formula [\textit{A. Weil}, Acta Math. 113, 1-87 (1965; Zbl 0161.023)] for the n-fold cover \(^{\sim}_ r({\mathbb{A}})\) of \(GL_ r({\mathbb{A}})\), where \({\mathbb{A}}\) is the ring of adèles of an A- field k containing all the n-th-roots of unity. Here n is a positive integer coprime to the characteristic of k. For \(^{\sim}_ 2({\mathbb{A}})\) and \(n=2\), the formula proves that the classical theta-functions are in fact residues of Eisenstein series [\textit{T. Kubota}, On automorphic functions and the reciprocity law in a number field (1969; Zbl 0231.10017); and \textit{S. Gelbart} and \textit{P. Sally}, Proc. Natl. Acad. Sci. USA 72, 1406-1410 (1975; Zbl 0303.22010)]. In fact, generalized theta-functions are certain functions on \(GL_ r(k)^*\setminus^{\sim}_ r({\mathbb{A}})\). Using the language of group representations they may be considered as certain constituents of \(L^ 2(GL_ r(k)^*\setminus GL_ r({\mathbb{A}})).\) More precisely, at every place v of k, there is always a reducible principal series representation of \(^{\sim}_ r(k_ v)\) which has a unique distinguished quotient (in the terminology of \textit{S. Gelbart} and \textit{I. I. Piatetski-Shapiro}, Invent Math. 59, 145-188 (1980; Zbl 0426.10027), i.e. it has a unique Whittaker model). This is in fact only true if \(r=n\) or n-1. The global representation which has these distinguished representations as its local components is then the generalized theta form, and the main result of this paper, Theorem II.2.2, proves that it is in fact the residue of an Eisenstein series, and consequently is automorphic. When \(r=2\), \(n=3\), and \(k={\mathbb{Q}}(\sqrt{-3})\), a similar result has been proved by the second author [J. Reine Angew. Math. 296, 125-161 (1977; Zbl 0358.10011) and 217-220 (1977; Zbl 0358.10012)] using classical language; and \textit{P. Deligne} [Sémin. Bourbaki, Exp. No.539, Lect. Notes Math. 770, 244-277 (1980; Zbl 0433.10018)], using group representations.…
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11R56
zbMATH DE Number: 3891469
zbMATH Keywords: dimension formula
zbMATH Keywords: Siegel-Weil formula
zbMATH Keywords: theta-functions
zbMATH Keywords: residues of Eisenstein series
zbMATH Keywords: generalized theta-functions
zbMATH Keywords: representation theory
zbMATH Keywords: metaplectic groups
zbMATH Keywords: space of Whittaker functionals
zbMATH Keywords: admissible representation
author: David Kazhdan
author: Samuel James Patterson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3891469