Shalom, Yehuda Explicit Kazhdan constants for representations of semisimple and arithmetic groups. (English) Zbl 0966.22004 Ann. Inst. Fourier 50, No. 3, 833-863 (2000). 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. Reviewer: Udai Tewari (Kanpur) Cited in 27 Documents MSC: 22D10 Unitary representations of locally compact groups 22E35 Analysis on \(p\)-adic Lie groups 22D30 Induced representations for locally compact groups 22E46 Semisimple Lie groups and their representations 43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc. Keywords:semisimple groups; arithmetic groups; lattices; property \((T)\); Kazhdan constants; topological group; Kazhdan set × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [Ba] , Unitary dual of Sp(n, 1), n ≥ 2, Duke Math. Journal, 48 (1981), 549-583. · Zbl 0496.22019 [2] [BaSw] and , On L2-cohomology and property (T) for automorphism groups of polyhedral cell complexes, GAFA, 7 (1997), 615-645. · Zbl 0897.22007 [3] [BB] and , The unitary spectrum for real rank one groups, Invent. Math., 72 (1983), 27-55. · Zbl 0561.22009 [4] [Be1] , On uniqueness of invariant means, Proc. 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