Weyl groups and rigidity of von Neumann algebras

Let $G$ be a noncompact semisimple algebraic group with trivial center, $S<G$ a maximal split torus, $H<G$ the centralizer of $S$ in $G$ and $\mathrm{\Gamma }<G$ an irreducible lattice. Consider the group measure space von Neumann algebra $\mathcal{M}=\mathrm{L}(\mathrm{\Gamma }\curvearrowright G\mathrm{/}H)$ associated with the nonsingular action $\mathrm{\Gamma }\curvearrowright G\mathrm{/}H$ and regard the group von Neumann algebra $M=\mathrm{L}(\mathrm{\Gamma })$ as a von Neumann subalgebra $M\subset \mathcal{M}$. We show that the group ${\mathrm{Aut}}_M(\mathcal{M})$ of all unital normal $\ast$-automorphisms of $\mathcal{M}$ acting identically on $M$ is isomorphic to the Weyl group ${\mathcal{W}}_G$ of the semisimple algebraic group $G$. Our main theorem is a noncommutative analogue of a rigidity result of Bader-Furman-Gorodnik-Weiss for group actions on algebraic homogeneous spaces and moreover gives new insight towards Connes’ rigidity conjecture for higher rank lattices.