Let G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let K° be the space of functions on the Poisson boundary Ω, 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° , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on K° and topologically irreducible on K° endowed with the weak topology.
Special series of unitary representations of groups acting on homogeneous trees
MANTERO, ANNA MARIA;ZAPPA, ANNA
1986-01-01
Abstract
Let G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let K° be the space of functions on the Poisson boundary Ω, 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° , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on K° and topologically irreducible on K° endowed with the weak topology.
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