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On diameters of orbits of compact groups in unitary representations. (English) Zbl 0853.22004

Let \(G\) be a locally compact group, and let \(S\) be a compact generating set of \(G\). The Kazhdan constant \(\kappa (G,S)\) of \((G,S)\) is defined as follows. Define first for a unitary representation \(\pi\) of \(G\) the constant \[ \kappa (\pi, G,S) = \inf_{|\xi |= 1} \sup_{x \in S} \bigl |\pi (x) \xi - \xi \bigr |, \] and then define \(\kappa (G,S) = \inf_\pi \kappa (\pi,G,S)\), where the infimum is taken over all unitary representations \(\pi\) of \(G \) without nonzero fixed vector. The group \(G\) has Kazhdan property \((T)\) if and only if \(\kappa (G,S) > 0\) for some (and hence for any) compact generating set \(S\). It is an interesting problem (attributed to J.-P. Serre in the monograph on property \((T)\) by P. de la Harpe and A. Valette [Astérisque 175, Paris: Soc. Math. France (1989; Zbl 0759.22001)]) to explicitly compute such constants.
In the article under review, the authors consider the case where \(G \) is compact and take \(G\) itself as a generating set. They show that \[ \begin{aligned} \text{(i)} \quad & \kappa (G,G) = \sqrt {{2n \over n - 1}} \quad \text{if } G \text{ is a finite group of order } n, \text{ and} \\ \text{(ii)} \quad & \kappa (G,G) = \sqrt 2 \quad \text{ if } G \text{ is a compact infinite group}.\end{aligned} \] Actually, they prove that \(\kappa (\pi, G, G) \geq \sqrt {{2n \over n - 1}}\) (resp. \(\geq \sqrt 2)\) for any representation \(\pi\) without fixed vector and that \(\kappa (\lambda^0,G,G) = \sqrt {{2n \over n - 1}}\) (resp. \(= \sqrt 2)\) for the restriction \(\lambda^0\) of the regular representation of \(G\) to the orthogonal of the constants.
Reviewer: M.B.Bekka (Metz)

MSC:

22C05 Compact groups
22D10 Unitary representations of locally compact groups
28D15 General groups of measure-preserving transformations

Citations:

Zbl 0759.22001
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