Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings

Abstract

We obtain a complete characterisation of factorial multiparameter Hecke von Neumann algebras associated with right-angled Coxeter groups. Considering their ${\ell }^p$-convolution algebra analogues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier for group Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us to draw conclusions on spherical representation theory of groups acting on right-angled buildings, which are in strong contrast to behaviour of spherical representations in the affine case. We also investigate certain graph product representations of right-angled Coxeter groups and note that our von Neumann algebraic structure results show that these are finite factor representations. Further classifying a suitable family of them up to unitary equivalence allows us to reveal high-dimensional Euclidean subspaces of the space of extremal characters of right-angled Coxeter groups.

arXiv commentv6: final version accepted for publication in J. Math. Pures Appl. v5: added details and corrections v4: correction a mistake in Lemma 3.2 through a major revision of Section 3. Theorem A takes into account additional one-dimensional direct summands v3: fixed minor inaccuracies; v2: added references