Automorphic Green functions associated with the secondary spherical functions
type: Item
instance of: scholarly article
title: Automorphic Green functions associated with the secondary spherical functions
zbMATH Open document ID: 1044.11033
author: Takayuki Oda
author: Masao Tsuzuki
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2004-03-07T00:00:00Z
review text: Let \(G\) be a connected semisimple algebraic group of Hermitian type defined over \(\mathbb Q\), and let \(K\) be a maximal compact subgroup of \(G_\mathbb R\). Then \(\mathcal D = G_\mathbb R /K\) is a bounded symmetric domain, and an arithmetic subgroup \(\Gamma\) of \(G\) determines the corresponding arithmetic variety \(\Gamma \backslash \mathcal D\). Let \(H\) be a reductive subgroup of \(G\) such that \(H_\mathbb R \cap K\) is a maximal compact subgroup of \(H_\mathbb R\) and \(\mathcal D_0 = H_\mathbb R \cap K \backslash H_\mathbb R\) is a subdomain of \(\mathcal D\) of complex codimension one. In this paper the authors introduce a spherical function \(\phi_s\) on \(G\) for \(s \in \mathbb C\) and define a Poincaré series \(\mathcal P_s\) on \(\Gamma \backslash \mathcal D\) as an infinite series involving \(\phi_s\). They prove that \(\mathcal P_s\) is meromorphic and is a fundamental solution of a Poisson equation. They also give a proof of the meromorphic continuation of \(\mathcal P_s\) with respect to \(s\) and show that \(\mathcal P_s\) satisfies an expected property of a Green current of logarithmic type.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 2051909
zbMATH Keywords: modular varieties
zbMATH Keywords: spherical functions
zbMATH Keywords: Poincaré series
zbMATH Keywords: Green functions
zbMATH Keywords: automorphic forms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145476077
OpenAlex ID: W2001525506
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DOI: 10.2977/PRIMS/1145476077
description: scientific article; zbMATH DE number 2051909