Monotony of Certain Free Group Representations

Let Γ be a free nonabelian group and let Ω be its boundary. Let πh be one of the unitary representations of Γ introduced earlier by the authors in (1996, Duke Math. J.82, 381–436). By its definition πh acts on L2(Ω, dνh) for a certain measure νh. This gives a boundary realization of πh in a sense we make precise. We show that πh does not have any other boundary realizations and simultaneously provide a new proof that πh is irreducible.