Explicit Kazhdan constants for representations of semisimple and arithmetic groups
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.
scientific article; zbMATH DE number 1478806
scientific article