Almost algebraic actions of algebraic groups and applications to algebraic representations

Abstract

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 [BF13].

Super-rigidity and non-linearity for lattices in products

“…As the connected group H 0 has no Zariski closed subgroups of finite index, we conclude that H 0 is normal in G. By the assumption that Γ is infinite we get that H 0 is non-trivial. By the simplicity of G we conclude that H 0 = G. In particular H = G. Thus indeed, Γ is Zariski dense in G. [2], as the latter considers merely locally compact groups. We thank the anonymous referee for spotting this gap, which is closed by the proof below.…”

“…(1) For every field K, integer d and a group homomorphism φ : Γ → GL d (K), φ(Γ) is solvable-by-locally finite. (2) For every finite index subgroup Γ ′ < Γ, complete field with an absolute value k, connected adjoint k-simple algebraic group G and Zariski dense group homomorphism ρ :…”

“…. Furthermore, if Γ is assumed to be finitely generated, the class of fields considered in (2) could be taken to be the class of local fields.…”

“…We denote by Gr d (E) the Grassmannian of d-dimensional k-subspaces of E and consider the map S d (E) → Gr d (E) taking a seminorm to its kernel. This map is measurable by [2,Proposition 5.4]. Pushing forward the measure µ we obtain an R-invariant measure ν on Gr d (E).…”

“…It follows that R is compact in G(k) and, in particular, it is bounded. 2 Recall that through out this paper we use the convention that a topological group is amenable if every continuous action of it on a compact space admits an invariant measure (note that various notions of amenability which are equivalent for lcsc groups need not coincide in general).…”