On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices

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 $\mathrm{\Gamma }<G$ be an irreducible lattice, let $P<G$ be a minimal parabolic subgroup, and consider the crossed product $L^{\mathrm{\infty }}(G\mathrm{/}P,{\nu }_P)⋊\mathrm{\Gamma }$. We prove that every $\mathrm{\Gamma }$-invariant von Neumann subalgebra of $L^{\mathrm{\infty }}(G\mathrm{/}P,{\nu }_P)⋊\mathrm{\Gamma }$ is of the form $L^{\mathrm{\infty }}(G\mathrm{/}Q,{\nu }_Q)⋊\mathrm{\Lambda }$, where $P\le Q\le G$ and $\mathrm{\Lambda }⊲\mathrm{\Gamma }$. This confirms a conjecture of Amrutam–Hartman.