On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices

Abstract

Let $G$ be a real connected semisimple Lie group with trivial center, no non-trivial compact factors, and all simple factors of real rank at least two. Let $Γ<G$ be an irreducible lattice, let $P<G$ be a minimal parabolic subgroup, and consider the crossed product $L^\infty(G/P,ν_P)\rtimes Γ$. We prove that every $Γ$-invariant von Neumann subalgebra of $L^\infty(G/P,ν_P)\rtimes Γ$ is of the form $L^\infty(G/Q,ν_Q)\rtimes Λ$, where $P\leq Q\leq G$ and $Λ\lhdΓ$. This confirms a conjecture of Amrutam--Hartman.