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 $\Gamma <G$ be an irreducible lattice and $P<G$ be a minimal parabolic subgroup. Amrutam--Hartman conjectured that every $\Gamma$-invariant von Neumann subalgebra $M\subset L^{\infty }(G/P,{\nu }_P)⋊\Gamma$ is of the form $L^{\infty }(G/Q,{\nu }_Q)⋊\Lambda$. We prove this classification whenever either $M\cap L^{\infty }(G/P,{\nu }_P)\ne \mathbb{C}1$ or $M\cap L(\Gamma )\ne \mathbb{C}1$, thereby reducing the Amrutam--Hartman conjecture to the extreme case $M\cap L^{\infty }(G/P,{\nu }_P)=M\cap L(\Gamma )=\mathbb{C}1$.

v3, 13 pages