Explicit Kazhdan constants for representations of semisimple and arithmetic groups
type: Item
instance of: scholarly article
title: Explicit Kazhdan constants for representations of semisimple and arithmetic groups
zbMATH Open document ID: 0966.22004
author: Yehuda Shalom
publication date: 2000-07-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75440
review text: Let \(G\) be a topological group and \({\mathcal F}\) a family of continuous unitary \(G\)-representations. \({\mathcal F}\) is said to be isolated from the trivial representation, if there is a compact subset \(Q\) of \(G\) and \(\varepsilon> 0\) such that \(\forall(n,{\mathcal H})\in{\mathcal F}\) there is no nonzero vector \(v\in{\mathcal H}\) which is \((Q,\varepsilon)\)-invariant, i.e. for which \[ \|\pi(g)v- v\|< \varepsilon\|v\|,\quad \forall g\in Q. \] In this case \(Q\) is called a Kazhdan set and \(\varepsilon\) a Kazhdan constant (for \(Q\)) for the family \({\mathcal F}\). If there are a compact set \(Q\) and \(\varepsilon> 0\), which form a Kazhdan set and constant for the family of all continuous unitary \(G\)-representations which do not contain a nonzero \(G\)-invariant vector, then \(G\) is said to have property (T). If there is no \(\varepsilon'>\varepsilon\) satisfying the above condition, then \(\varepsilon\) is said to be the best Kazhdan constant for \(Q\). Kazhdan sets and constants for various classes of semisimple Lie groups have been studied by several authors. In this paper the author describes explicit Kazhdan constants for every group of rational points of a semisimple, almost \(k\)-simple algebraic group with property (T), over any locally compact nondiscrete field \(k\), and consequently, for all lattices in such groups as well. Moreover, for the algebraic groups, it is shown that the Kazhdan constants obtained are best possible.
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 1478806
zbMATH Keywords: semisimple groups
zbMATH Keywords: arithmetic groups
zbMATH Keywords: lattices
zbMATH Keywords: property \((T)\)
zbMATH Keywords: Kazhdan constants
zbMATH Keywords: topological group
zbMATH Keywords: Kazhdan set
MaRDI profile type: Publication
OpenAlex ID: W2001282789
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Explicit Kazhdan constants for representations of semisimple and arithmetic groups (cites work)
cites work: Property (T) and Kazhdan constants for discrete groups

Explicit Kazhdan constants for representations of semisimple and arithmetic groups (DOI)
DOI: 10.5802/AIF.1775

Explicit Kazhdan constants for representations of semisimple and arithmetic groups (published in)
published in: Annales de l’institut Fourier

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