Let be an algebraic group over a complete separable valued field . We discuss the dynamics of the -action on spaces of probability measures on algebraic -varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis–Zimmer super-rigidity phenomenon .
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Uri Bader, Bruno Duchesne, Jean Lécureux, Almost algebraic actions of algebraic groups and applications to algebraic representations. Groups Geom. Dyn. 11 (2017), no. 2, pp. 705–738DOI 10.4171/GGD/413