The resolvent trace formula for rank one Lie groups
type: Item
instance of: scholarly article
title: The resolvent trace formula for rank one Lie groups
zbMATH Open document ID: 1026.22010
author: Yasuro Gon
author: Masao Tsuzuki
published in: The Asian Journal of Mathematics
publication date: 2003-11-27T00:00:00Z
review text: Let \(G\) be a connected semisimple Lie group of real rank one with finite center and \(K\) be its maximal compact subgroup. Then \(X = G/K\) is a Riemannian symmetric space of non-compact type. Introduce, following \textit{R. J. Miatello} and \textit{N. R. Wallach} [J. Differ. Geom. 36, 663-698 (1992; Zbl 0766.53044)], a family of bi-\(K\)-invariant functions \(Q_s\), \(s \in C\), on \(G\) and define then a function \(Q_{r,s}\) on \(G\) via the \(r\)-fold convolution of \(Q_s\). A formula of \(Q_{r,s}\) in terms of a derivative of the hypergeometric series is derived and the integral of the Poincaré series is computed via partitioning it into the local contributions for \(\Gamma\)-conjugacy classes. Consequently, the integral is evaluated by means of the logarithmic derivative of the Selberg zeta function for \(\Gamma \setminus X\) arriving thus at the resolvent trace formula.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1925758
zbMATH Keywords: Lie group of rank one spherical functions
zbMATH Keywords: Miatello-Wallach's functional
zbMATH Keywords: resolvent trace formula
zbMATH Keywords: Selberg zeta function
reviewed by: Eugene Kryachko
MaRDI profile type: Publication
Wikidata QID: Q115206241
full work available at URL: ajm.2002.v6.n2.a2
OpenAlex ID: W1607798493
DOI: 10.4310/AJM.2002.V6.N2.A2
description: scientific article; zbMATH DE number 1925758