Opérateurs invariants sur certains immeubles affines de rang 2

Ferdaous Kellil (1) , Guy Rousseau (2)

1 Département de Mathématiques [Monastir]
2 IECN - Institut Élie Cartan de Nancy

We consider a building $\Delta$ of type $\widetilde A_2$ or $\widetilde B_2$, different subsets $S'$ of vertices in $\Delta$ and different automorphism groups $G$, very strongly transitive on $\Delta$. We prove that the algebra of $G-$invariant operators acting on the space of functions on $S'$ is often not commutative (contrarily to the classical results). In some cases we describe its structure, determine its radial eigenfunctions and deduce that the Helgason conjecture (about the Poisson transform) is not verified in this context.

Mots clés: buildings; Laplace operators; Poisson kernel; convolution algebra

Domaines: Théorie des groupes [math.GR]

Ferdaous Kellil, Guy Rousseau. Opérateurs invariants sur certains immeubles affines de rang 2. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2007, XVI (3), pp.591-610. ⟨10.5802/afst.1160⟩. ⟨hal-00094818⟩