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 $\mathcal{ℳ}=L(\Gamma \curvearrowright G∕H)$ associated with the nonsingular action $\Gamma \curvearrowright G∕H$ and regard the group von Neumann algebra $M=L(\Gamma )$ as a von Neumann subalgebra $M\subset \mathcal{ℳ}$. We show that the group ${\mathrm{Aut}<!--FUNCTION\; APPLICATION-->}_M(\mathcal{ℳ})$ 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 gives new insight towards Connes’ rigidity conjecture for higher rank lattices.

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