Weyl groups and rigidity of von Neumann algebras

TL;DR

This paper establishes a deep connection between automorphisms of certain von Neumann algebras and Weyl groups of semisimple algebraic groups, providing new insights into rigidity phenomena and Connes' conjecture.

Contribution

It proves that the automorphism group of a von Neumann algebra associated with a lattice action is isomorphic to the Weyl group, extending rigidity results to a noncommutative setting.

Findings

Automorphism group of the von Neumann algebra is isomorphic to the Weyl group.

Provides a noncommutative analogue of a known rigidity result.

Offers new perspectives on Connes' rigidity conjecture for higher rank lattices.

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.