The trace formula for Hecke operators over rank one lattices
type: Item
instance of: scholarly article
title: The trace formula for Hecke operators over rank one lattices
zbMATH Open document ID: 0699.22016
DOI: 10.1016/0022-1236(89)90104-3
author: Werner Hoffmann
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let \(G\) be a real reductive group and let \(\Gamma \subset G\) be a lattice of rank one, in the sense of [OW] [\textit{M. Osborne} and \textit{G. Warner}, J. Reine Angew. Math. 324, 1--113 (1981; Zbl 0469.22007)]. Let \(\pi\) denote the left regular representation of \(G\) on \(L^ 2(G/\Gamma)\) and let \(L^ 2_ d(G/\Gamma)\) denote the closure of the sum of the \(G\)-invariant irreducible subspaces of \(L^ 2(G/\Gamma)\). If \(\alpha \in C_ c^{\infty}(G)\) is \(K\)-finite and \(T\) is a Hecke operator, the operator \(\pi(\alpha)T\) is trace-class, by a result of Donelly (recently W. Müller has proved this without restriction on the rank). The purpose of the present paper is to arrive at an explicit expression for its trace, under an assumption analogous to that in [OW], in the case of \(\pi(\alpha)\). The trace is given in terms of (weighted) orbital integrals and the usual intertwining and residual terms. The exposition is self- contained and the proofs follow the lines of those in [OW], in the case of \(\pi (\alpha)\).
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 4146292
zbMATH Keywords: real reductive group
zbMATH Keywords: lattice of rank one
zbMATH Keywords: left regular representation
zbMATH Keywords: \(K\)-finiteness
zbMATH Keywords: Hecke operator
zbMATH Keywords: trace-class
zbMATH Keywords: orbital integrals
reviewed by: R. J. Miatello
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90104-3
OpenAlex ID: W1975874833
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description: scientific article; zbMATH DE number 4146292