Sous-groupes paraboliques et représentations de groupes branchés

Abstract

Let G be a branch group (in the sense of [9]) acting on a tree T. A parabolic subgroup P is the stabilizer of an infinite geodesic ray in T. We denote by ρ 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-dimensional representations ρ G/P n+1 ⊖ρ G/P n , where P n is the stabilizer of a level- n vertex in T. For a few concrete examples (notably a virtually torsion-free branch group), we completely split ρ G/P n in irreducible components; (G,P n) and (G,P) are Gelfand pairs (producing abelian Hecke algebra).