Sous-groupes paraboliques et representations de groupes branches

Details Sous-groupes paraboliques et representations de groupes branches Sous-groupes paraboliques et representations de groupes branches

2000-12-18

arxiv.org/abs/math/0012175v1

C. R. Acad. Sci. Paris, Ser. Math. 332 (2001), no. 9, 789-794

Mathematics

Group Theory

Short note, in french, summing up math.GR/9911206, to appear in C. R. Acad. Sci. Paris

Scientific paper

10.1016/S0764-4442(01)01946-2

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.

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