Almost algebraic actions of algebraic groups and applications to algebraic representations

Uri Bader (1) , Bruno Duchesne (2) , Jean Lécureux (3)

1 TECHNION - Department of Mathematics
2 IECL - Institut Élie Cartan de Lorraine
3 LMO - Laboratoire de Mathématiques d'Orsay

Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-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 [2].

Mots clés: Complete separable valued fields; probability measures on algebraic varieties; algebraic representations of amenable ergodic actions; Margulis–Zimmer super-rigidity

Domaines: Géométrie différentielle [math.DG]

Uri Bader, Bruno Duchesne, Jean Lécureux. Almost algebraic actions of algebraic groups and applications to algebraic representations. Groups Geom. Dyn, 2017, 11 (2), pp.705 - 738. ⟨10.4171/GGD/413⟩. ⟨hal-01230558v2⟩