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 and $P<G$ be a minimal parabolic subgroup. Amrutam--Hartman conjectured that every $Γ$-invariant von Neumann subalgebra $M\subset L^\infty(G/P,ν_P)\rtimes Γ$ is of the form $L^\infty(G/Q,ν_Q)\rtimesΛ$. We prove this classification whenever either $M\cap L^\infty(G/P,ν_P)\neq\mathbb{C}1$ or $M\cap L(Γ)\neq\mathbb{C}1$, thereby reducing the Amrutam--Hartman conjecture to the extreme case $M\cap L^\infty(G/P,ν_P)=M\cap L(Γ)=\mathbb{C}1$.