Boundary unitary representations - right-angled hyperbolic buildings

TL;DR

This paper investigates the irreducibility of boundary unitary representations of groups acting on right-angled hyperbolic buildings, linking them to Hecke algebra representations and establishing their irreducibility.

Contribution

It introduces a novel approach connecting boundary representations of hyperbolic buildings to Hecke algebra representations, proving their irreducibility.

Findings

Boundary unitary representations are irreducible.

Associated Hecke algebra representations are irreducible.

Establishes a new link between hyperbolic building actions and algebraic structures.

Abstract

We study the unitary boundary representation of a strongly transitive group acting on a right-angled hyperbolic building. We show its irreducibility. We do so by associating to such a representation a representation of a certain Hecke algebra, which is a deformation of the classical representation of a hyperbolic reflection group. We show that the associated Hecke algebra representation is irreducible.