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Kazhdan constants and matrix coefficients of \(\text{Sp}(n,\mathbb{R})\). (English) Zbl 1020.22004

A locally compact group \(G\) is said to have Kazhdan’s property \(T\) if there exists a compact subset \(Q\) and \(\varepsilon> 0\) such that every unitary representation \(\pi\) which has a \((Q,\varepsilon)\) invariant vector \(\xi\in H_\pi\), i.e. a vector \(\xi\in H_\pi\) such that \(\|\pi(g)\xi- \xi\|< \varepsilon\|\xi\|\) for all \(g\in Q\), has in fact a nonzero invariant vector. If such a \((Q,\varepsilon)\) for a group exists it is called a Kazhdan pair. The author proves that the group \(\text{Sp}(n,\mathbb{R})\), \(n\geq 2\), satisfies Kazhdan’s property \(T\) and determines an explicit Kazhdan pair \((Q,\varepsilon)\).

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22D10 Unitary representations of locally compact groups
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