Kazhdan constants for conjugacy classes of compact groups.
Let \(G\) be a locally compact group and \(\pi\) a unitary representation of \(G\) in the Hilbert space of \({\mathcal H}_\pi\). Associated to \(\pi\) and any compact subset \(Q\) of \(G\) is a so-called Kazhdan constant defined by \[ K(\pi,G,Q)= \inf\Biggl\{\sup_{x\in Q}\|\pi(x) \xi- \xi\|:\xi\in{\mathcal H}_\pi,\|\xi\|= 1\Biggr\}. \] Let \(R\) denote the set of equivalence classes of representations of \(G\) on separable Hilbert spaces not containing the trivial representation. Varying \(\pi\) through \(R\) and taking the infimum, one obtains the absolute Kazhdan constant relative to \(Q\), \(K(G, Q)\). Then \(G\) has Kazhdan's property (T) exactly when \(K(G, Q)> 0\) for some compact subset \(Q\) of \(G\). These Kazhdan constants may be viewed as a quantitative version of property (T) and have proved to play an important role in various applications. Although compact groups trivially share property (T), computing Kazhdan constants for compact groups in highly non-trivial. The paper under review is devoted to calculating \(K(G, Q)\) where \(Q\) is a conjugacy class of a compact group \(G\). The main result is an explicit formula for \(K(G, Q)\) in terms of the degrees of non-trivial irreducible characters \(\chi\) of \(G\) and the values \(\chi(Q)\) (Theorem 1.1). This is applied to the special unitary group \(\text{SU}(n)\) and certain conjugacy classes \(Q\) to obtain lower estimates for \(K(\text{SU}(n), Q)\) and precise values when \(n= 2\). Another application concerns symmetric groups.
scientific article; zbMATH DE number 2037077
scientific article