## 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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