Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T)

Shalom, Yehuda

doi:10.1090/S0002-9947-99-02363-6

Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T)
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Trans. Amer. Math. Soc. 351 (1999), 3387-3412 Request permission

Abstract:

Let $k$ be any locally compact non-discrete field. We show that finite invariant measures for $k$-algebraic actions are obtained only via actions of compact groups. This extends both Borel’s density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for $k$-algebraic actions, finitely additive finite invariant measures are obtained only via actions of amenable groups. This gives a new criterion for Zariski density of subgroups and is shown to have representation theoretic applications. The main one is to Kazhdan’s property $(T)$ for algebraic groups, which we investigate and strengthen.

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