Kazhdan constants and the dual space topology
Let \(G\) be a locally compact group. For a compact subset \(S\) of \(G\) and for \(\pi\in\widehat {G}\) define the Kazhdan constant of \(S\) and \(\pi\) by \[ K_ S(\pi)= \inf_{\xi\in {\mathcal H}_ \pi^ 1} (\max_{x\in S} \|\pi(x)\xi-\xi\|). \] In this paper the authors study the continuity of the function \(\pi\to K_ S(\pi)\). They prove for an almost connected \(G\) that \(K_ S\) is continuous on the whole space \(\widehat {G}\) for some compact subset \(S\) which generates \(G\) if and only if \(G\) contains a compact normal subgroup \(K\) such that \(G/K\) is abelian. Let now \(G\) be a separable locally compact group and \(H\) a closed subgroup of \(G\) such that \(G/H\) is compact. Suppose that \(\Sigma\) is a set of representations of \(H\) which act in the same finite dimensional Hilbert space \({\mathcal H}\). Then \(\sigma\to K_ S(\text{ind}_ H^ G \sigma)\) is a continuous function on \(\Sigma\). The authors apply this result to Moore groups and to motion groups where they show that \(K_ S\) is continuous on large open subsets of \(\widehat {G}\). Finally they show for acceptable connected semi-simple groups, that \(K_ S\) is continuous on the set of irreducible principal series representations.
scientific article; zbMATH DE number 126347
scientific article