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Sous-groupes paraboliques et représentations de groupes branchés

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Published in Comptes Rendus de l'Académie des Sciences. I, Mathematics. 2001, vol. 332, no. 9, p. 789-794
Abstract 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 $ ho_{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 $ ho_{G/P_{n+1}}ominus ho_{G/P_n}$, where P_n is the stabiliser of a level-n vertex in T. For a few concrete examples, we completely split $ ho_{G/P_n}$ in irreducible components. $(G,P_n)$ and $(G,P)$ are Gelfand pairs, whence new occurrences of abelian Hecke algebra.
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Deposited on : 2010-11-08

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