A relative trace identity between \(\text{GL}_{2n}\) and \(\widetilde{\text{Sp}}_n\)
type: Item
instance of: scholarly article
title: A relative trace identity between \(\text{GL}_{2n}\) and \(\widetilde{\text{Sp}}_n\)
zbMATH Open document ID: 1259.11054
DOI: 10.1215/00127094-2010-012
author: Zhengyu Mao
author: Steve Rallis
published in: Duke Mathematical Journal
publication date: 2010-05-11T00:00:00Z
full work available at URL: GL-Mp.pdf
review text: The paper under review studies the functorial lift from the metaplectic group \(\widetilde{\text{Sp}}_n\) (sometimes denoted as \(\widetilde{\text{Sp}}_{2n}\) in the literature) to \(\text{GL}_{2n}\) via a comparison of relative trace formulas. Such functorial lifts have been studied from different angles, eg. via the Arthur-Selberg trace formula. One of the advantage of the relative trace formulas is that it yields information about the central L-value \(L(1/2, \Pi)\), which generalizes Waldspurger's celebrated work for the case \(n=1\). To be precise, let \(k\) be a number field, \(\mathbb{A}\) be its ring of adèles, and \(G\) be a connected reductive \(k\)-group. The relative trace formula associated to a datum \((H_1, \chi_1; H_2, \chi_2)\) concerns the distribution \(f \mapsto I_G(f: H_1, \chi_1; H_2, \chi_2)\), that is, the integration of the kernel function \(K_f(x,y)\) along \(H_1(k) \backslash H_1(\mathbb{A}) \times H_2(k) \backslash H_2(\mathbb{A})\). Here \(f\) lies in the Schwartz space \(\mathcal{S}(G(\mathbb{A}))\); for \(i=1,2\), \(H_i\) is a closed subgroup of \(G\) and \(\chi_i\) is an automorphic character of \(H_i(\mathbb{A})\). We will also need the generalization in which \(\chi_i\) is multiplied by a certain theta function. In the cases encountered in this paper, such integrals are absolutely convergent. The main result (Theorem 1.1) is a relative trace identity of the form \[ I_{\text{GL}_{2n}}(f: \text{GL}_n \times \text{GL}_n, 1; N, \theta) = I_{\widetilde{\text{Sp}}_n}(\tilde{f}: N', {\theta'}^{-1}; N', \theta') \] for factorizable test function \(f\) and a certain ``transfer map'' \(f \mapsto \tilde{f}\). Here \(N\), \(N'\) are appropriate maximal unipotent subgroups and \(\theta\), \(\theta'\) are non-degenerate characters. Such an identity is expected to yield some relation between \(|W(\tilde{\varphi})|^2\) (Whittaker functional on \(\widetilde{\text{Sp}}_n\)) and \(W(\varphi)\) (Whittaker functional on \(\text{GL}_{2n}\)) times the \(\…
reviewed by: Wen-Wei Li
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 5706942
zbMATH Keywords: relative trace formula
zbMATH Keywords: period
zbMATH Keywords: automorphic descent
MaRDI profile type: Publication
OpenAlex ID: W2036888200
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description: scientific article; zbMATH DE number 5706942