Monotony of certain free group representations (Q5927504)

Let \(\Gamma\) be a nonabelian free group with finitely many generators. Using a complex function \(h\) on the set of generators, the authors introduced a unitary representation \(\pi_h\) of \(\Gamma\) on \(L^2 (\Omega, \nu_h)\) for a certain measure \(\nu_h\) on the boundary \(\Omega\) of \(\Gamma\) in [Duke Math. J. 82, 381-436 (1996; Zbl 0851.22005)]. In the paper under review it is shown that such representations \(\pi_h\) admit unique such boundary realizations, and a new proof is given for the fact that \(\pi_h\) is irreducible.