Special series of unitary representations of groups acting on homogeneous trees

Let G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let K\ub0 be the space of functions on the Poisson boundary \u3a9, of zero mean on K. When p is a prime, G is a discrete subgroup of PGL_2(Q_p) of finite covolume. The representations of the special series of PGL_2(Q_p), which are irreducible and unitary in an appropriate completion of K\ub0 , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on K\ub0 and topologically irreducible on K\ub0 endowed with the weak topology