Monotony of certain free group representations

Let Gamma be a ii-re nonabelian group and let Omega be its boundary. Let pi (h) be one of the unitary representations of Gamma introduced earlier by t he authors in ( 1996, Duke Math. J. 82. 381 436). By its definition pi (h) acts on L-2(Omega, dv(h)) for a certain measure v(h). This gives a boundary realization of pi (h) in a sense ne make precise. We show that pi (h) does not have any other boundary realizations and simultaneously provide a new proof th;lt pi (h) is irreducible. (C) 2001 Academic Press.