A Siegel-Weil identity for \(G_2\) and poles of \(L\)-functions
type: Item
instance of: scholarly article
title: A Siegel-Weil identity for \(G_2\) and poles of \(L\)-functions
zbMATH Open document ID: 0958.11035
author: David Ginzburg
author: Dihua Jiang
published in: Journal of Number Theory
publication date: 2001-04-11T00:00:00Z
review text: The authors study two examples of the relation among the following three properties: (1) \(\pi\), an irreducible, cuspidal, automorphic representation on \(G_{\mathbb{A}}\) is an endoscopic functorial lift from a (similar) representation \(\tau\) on \(H_{\mathbb{A}}\). (2) A certain (partial) \(L\)-function \(L^S(\pi, \rho, s)\) has a pole at \(s=1\). (3) A certain period is nontrivial on the space of \(\pi\). In the first example \(G= \text{SO}_7\) (split), \(\pi\) is generic and \(\rho\) is the second fundamental representation of \(\text{Sp}_6 (\mathbb{C})\) \((\dim\rho= 14)\), \(H= \text{SO}_3\times \text{SO}_3\) (split) and the period is taken along the subgroup \[ \left\{ u= \begin{pmatrix} 1 &x &y &z &*\\ && \vcenter{\vtop{=1em \line{\hss\hbox to 4.4em{\hrulefill}\hss} \line{\strut\hss\vrule\(\begin{matrix} g&0&t\\&1&0\\&&g^*\end{matrix}\)\vrule\hss\strut} \line{\hss\hbox to 4.4em{\hrulefill}\hss}}} &&\vcenter{\vtop{=1em\line{\strut\hss\(\begin{matrix}*\\*\\*\end{matrix}\)\hss\strut}}} \\ &&&&1 \end{pmatrix}\in \text{SO}_7\mid g\in \text{SL}_2 \right\} \] and the character \(u\mapsto \psi(y+ t_{11})\). The implication \((1) \Rightarrow (3)\) is clear. The authors conjecture that this is a special case of a more general set-up, where \(G= \text{SO}_{2n+1}\) (split), \(\rho\) is the second fundamental representation of \(\text{Sp}_{2n} (\mathbb{C})\), \(H= \text{SO}_3 \times \cdots\times \text{SO}_3\) (\(n\) copies of split \(\text{SO}_3)\), \(\tau= \bigotimes_{i=1}^n \tau_i\), where \(\tau_1,\dots, \tau_n\) are pairwise distinct cusp forms, and the period subgroups is the product of the above marked middle block (embedded in \(\text{SO}_{2n+1}\)) and the unipotent radical of the standard parabolic subgroup whose Levi part is isomorphic to \(\text{GL}_1^{n-2} \times \text{SO}_5\). The character is an ``extension'' of the above character by the standard nondegenerate character. In case \(n=2\), all three properties above are equivalent for generic \(\pi\) [\text…
reviewed by: David Soudry
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1466187
zbMATH Keywords: poles
zbMATH Keywords: periods
zbMATH Keywords: \(G_2\)
zbMATH Keywords: \(F_4\)
zbMATH Keywords: partial \(L\)-function
zbMATH Keywords: cuspidal automorphic representation
zbMATH Keywords: Rankin-Selberg integral
zbMATH Keywords: Siegel-Weil identity
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
full work available at URL: jnth.1999.2480
OpenAlex ID: W2084790502
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DOI: 10.1006/JNTH.1999.2480
description: scientific article; zbMATH DE number 1466187