Parabolic subgroups and representations of branch groups

Let G be a branch group (in the sense of [9]) acting on a tree J. A parabol ic subguop P is the stabilizer of an infinite geodesic ray in J. We denote by rho (G/P) the associated quasi-regular representation. If G is discrete, these representations are irreducible, but if G is profin ite, they split as a direct sum of finite-dimensional representations rho ( G/Pn+1) circle minus rho (G/Pn), where P-n, is the stabilizer of a level-n vertex in J. For a few concrete examples (notably a virtually torsion-free branch group) , we completely split rho (G)/P-n in irreducible componpnrs; (G, P-n) and ( G, P) are Gelfand pairs (producing abelian Hecke algebra). (C) 2001 Academi c des sciences/Editions scientifiques et medicales Elsevier SAS.