A regularized Siegel-Weil formula on \(\text{U}(2,2)\) and \(\text{U}(3)\)
type: Item
instance of: scholarly article
title: A regularized Siegel-Weil formula on \(\text{U}(2,2)\) and \(\text{U}(3)\)
zbMATH Open document ID: 0989.11025
DOI: 10.1215/S0012-7094-98-09416-9
author: Victor Tan
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: Since the work of Hecke it has been understood that theta series define automorphic forms on classical groups. They allow correspondences between certain pairs of groups, called dual reductive pairs by Howe, the initiator of the spectral approach; however, the idea behind such pairs goes back to the work of \textit{C. L. Siegel} [Ann. Math. (2) 36, 527-606 (1935; Zbl 0012.19703)], which was later reexamined by \textit{A. Weil} [Acta Math. 111, 143-211 (1964; Zbl 0203.03305); Acta Math. 113, 1-87 (1965; Zbl 0161.02304)]. The regularized Siegel-Weil formula for the dual reductive pair \((\text{Sp}(n),O(m))\) was established by \textit{S. Kudla} and \textit{S. Rallis} [Ann. Math. (2) 140, 1-80 (1994; Zbl 0818.11024)]. It relates an integral of theta functions to an Eisenstein series on \(\text{Sp}(n)\). One of the important applications of the regularized Siegel-Weil formula is in studying the relation between the existence of certain poles of the standard \(L\)-functions and the nonvanishing of the corresponding theta liftings. The classical form of this theorem is valid in the range \(m>2n+2\) when the integral on the orthogonal group converges and the Eisenstein series is regular at the point under consideration. Outside the range \(m>2n+2\), the theta integral may not converge and the Eisenstein series may have a pole. S. Kudla and S. Rallis [loc. cit.] have given a method to regularise the theta integral so that the resulting object becomes a meromorphic function. They then established a relation between the first term in the Laurent expansion of the Eisenstein series and the regularised theta integral. In another paper, \textit{S. Kudla, S. Rallis} and \textit{D. Soudry} [Invent. Math. 107, 483-541 (1992; Zbl 0776.11028)] proved a second term identity which involves a comparison between the next terms of the two Laurent series for \(\text{Sp}(2)\) and \(O(4)\). The aim of this paper is to establish the analogous first and second term identities for the dual reduc…
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 1425144
zbMATH Keywords: Weil representation
zbMATH Keywords: Siegel's mass formula
zbMATH Keywords: theta functions
zbMATH Keywords: regularization
zbMATH Keywords: Eisenstein series
zbMATH Keywords: analytic continuation of Eisenstein series
zbMATH Keywords: Siegel-Weil formula
zbMATH Keywords: automorphic forms
zbMATH Keywords: dual reductive pair
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09416-9
OpenAlex ID: W104949161
description: scientific article; zbMATH DE number 1425144