A regularized Siegel-Weil formula for unitary groups
type: Item
instance of: scholarly article
title: A regularized Siegel-Weil formula for unitary groups
zbMATH Open document ID: 1071.11022
author: Atsushi Ichino
published in: Mathematische Zeitschrift
publication date: 2005-02-11T00:00:00Z
review text: Let \(F\) be a totally real field with adele ring \(\mathbb A=\mathbb A_F\), and let \(E\) be a totally imaginary quadratic extension of \(F\). Let \((V,Q)\) be a non-degenerate hermitian space over \(E\) with Witt index \(r\). Let \(G=U(n,n)\) denote the quasi-split unitary group in \(2n\) variables, and \(H=U(V)\) denote the unitary group of \((V,Q)\). Then \((G,H)\) is a reductive dual pair. Let \(\psi\) be a fixed non-trivial additive character of \(\mathbb A/F\) and let \(\chi\) be a fixed character of \(A_E^\times/E^\times\) such that \(\chi_{\mathbb A^\times}=\varepsilon^m_{E| F}\), where \(\varepsilon_{E| F}\) is the quadratic character of \(\mathbb A^\times/F^\times\) corresponding to \(E/F\) by class field theory. Then \(G(\mathbb A)\times H(\mathbb A)\) acts on the Schwartz space \(\mathcal S(V^n(\mathbb A))\) by the Weil representation \(\omega_Q=\omega_{Q,\psi,\chi}\). Let \(P\) be the Siegel parabolic subgroup of \(G\), and let \(K_G\) be a fixed standard maximal compact subgroup of \(G(\mathbb A)\). Put \(| a(g)| _{\mathbb A_E}:=| \det(a)| _{\mathbb A_E}\) for \(g=pk\in G(\mathbb A)\), with \[ p=\left(\begin{matrix} a&*\cr 0_n&{}^t\overline{a}^{-1}\end{matrix}\right)\in P(\mathbb A),\;\;a\in\text{GL}_n(\mathbb A_E),\quad k\in K_G. \] For each \(\Phi\in\mathcal S(V^n(\mathbb A))\), one defines a holomorphic section \(f_\Phi^{(s)}\) of \(\text{Ind}_{P(\mathbb A)}^{G(\mathbb A)}(\chi\,| \det| ^s_{\mathbb A_E})\) by \[ f_\Phi^{(s)}(g):=| a(g)| ^{s-s_0}_{\mathbb A_E}\,\omega_Q(g)\,\Phi(0), \] where \(s_0=(m-n)/2\). Then the Siegel Eisenstein series \[ E(g,f_\Phi^{(s)}):=\sum_{\gamma\in P(F)\backslash G(F)} f_\Phi^{(s)}(\gamma g) \] is absolutely convergent if \(\Re(s)>n/2\), and has meromorphic continuation to the whole \(s\)-plane if \(\phi\) is \(K_G\)-finite. On the other side, one defines the theta function \[ \Theta(g,h;\Phi):=\sum_{x\in V^n(F)}\omega_Q(g,h)\,\Phi(x), \] for each \(\Phi\). Let \(dh\) be the Haar measure on \(H(\mathbb A)\) is normaliz…
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 2133480
zbMATH Keywords: Siegel-Weil formula
zbMATH Keywords: unitary group
zbMATH Keywords: Siegel Eisenstein series
zbMATH Keywords: theta function
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: s00209-003-0580-5
OpenAlex ID: W1994999180
DOI: 10.1007/S00209-003-0580-5
description: scientific article; zbMATH DE number 2133480