Weyl groups and rigidity of von Neumann algebras

Abstract

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 $\Gamma < G$ an irreducible lattice. Consider the group measure space von Neumann algebra $\mathscr M = \operatorname{L}(\Gamma \curvearrowright G/H)$ associated with the nonsingular action $\Gamma \curvearrowright G/H$ and regard the group von Neumann algebra $M = \operatorname{L}(\Gamma)$ as a von Neumann subalgebra $M \subset \mathscr M$. We show that the group $\operatorname{Aut}_M(\mathscr M)$ of all unital normal $\ast$-automorphisms of $\mathscr M$ acting identically on $M$ is isomorphic to the Weyl group $\mathscr W_G$ 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.