Sous-groupes paraboliques et representations de groupes branches

Let G be a branch group (as defined by Grigorchuk) acting on a tree T. A parabolic subgroup P is the stabiliser of an infinite geodesic ray in T. We denote by $\rho_{G/P}$ the associated quasi-regular representation. If G is discrete, these representations are irreducible, but if G is profinite, they split as a direct sum of finite-dimensionalrepresentations $\rho_{G/P_{n+1}}\ominus\rho_{G/P_n}$, where P_n is the stabiliser of a level-n vertex in T. For a few concrete examples, we completely split $\rho_{G/P_n}$ in irreducible components. $(G,P_n)$ and $(G,P)$ are Gelfand pairs, whence new occurrences of abelian Hecke algebra.