The trace class conjecture for arithmetic groups
type: Item
instance of: scholarly article
title: The trace class conjecture for arithmetic groups
zbMATH Open document ID: 0926.11034
author: Lizhen Ji
published in: Journal of Differential Geometry
publication date: 1998-10-28T00:00:00Z
review text: Let \(G\) a Lie group, real locus of a reductive algebraic group defined over \(\mathbb Q\) and \(\Gamma\) an arithmetic subgroup of \(G(\mathbb Q)\). The right \(G\)-module \({\mathcal H}=L^2(\Gamma\backslash G)\) decomposes as \({\mathcal H}={\mathcal H}_{d}\oplus {\mathcal H}_{c}\) where \({\mathcal H}_{d}=\oplus_{\pi\in\widehat G} m_\pi\pi\) is the smallest subspace of \({\mathcal H}\) containing all subrepresentations of \({\mathcal H}\) and \({\mathcal H}_c\) is a Hilbert continuous integral of irreducibles. \textit{A. Borel} and \textit{H. Garland} [Am. J. Math. 105, 309-335 (1983; Zbl 0572.22007)] proved the finiteness of each \(m_\pi\) and \textit{J. Arthur} [Ann. Math. (2) 100, 326-385 (1974; Zbl 0289.20027)] asked about the trace class property of the integrated representation \(R_d(\alpha)\) defined for \(\alpha\in{\mathcal C}^\infty_0(G)\) as the restriction on \({\mathcal H}_d\) of \[ R(\alpha)f(\Gamma g)=\int_G\alpha(h)f(\Gamma gh) dh, \quad g\in G . \] This trace class property enters in the Arthur trace formula, generalization to a wide class of groups of the Selberg trace formula [\textit{A. Selberg}, J. Indian Math. Soc., New Ser. 20, 47-87 (1956; Zbl 0072.08201)] proved originally for arithmetic lattices of \(G=SL(2,{\mathbb R})\). The trace class conjecture has been proved for \({\mathbb Q}\)-rank one groups firstly by \textit{H. Donnelly} [Mich. Math. J. 31, 349-357 (1984; Zbl 0591.58033)] and later by \textit{R. P. Langlands} [Isr. Math. Conf. Proc. 3, 111-125 (1990; Zbl 0729.11038)] and in the general case, but with the restriction of \(K\)-finiteness for \(\alpha\) (\(K\) is a maximal compact connected subgroup of \(G\)), by \textit{W. Müller} [Ann. Math. (2) 130, No. 3, 473-529 (1989; Zbl 0701.11019)]. In the paper under review, the author proves the conjecture for \(\alpha\) in the Harish-Chandra-Schwartz space of integrable functions without any \(K\)-finiteness restriction; the author studies the case of \(K\)-finite functions (related t…
reviewed by: Laurent Guillopé
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1187447
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: trace class conjecture
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Eisenstein series
zbMATH Keywords: scattering matrix
zbMATH Keywords: Weyl law
zbMATH Keywords: discrete spectrum
zbMATH Keywords: pseudo-Laplacian
zbMATH Keywords: automorphic forms
MaRDI profile type: Publication
Wikidata QID: Q115174797
full work available at URL: https://doi.org/10.4310/jdg/1214460610
OpenAlex ID: W1588155137
DOI: 10.4310/JDG/1214460610
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description: scientific article; zbMATH DE number 1187447