Opérateurs invariants sur un immeuble affine de type ${\tilde{B}}_n(n\ge 3)$

Abstract

We consider a building $\Delta$ of type \( \widetilde{B}_n~(n\geq3) \), different subsets ${\mathcal{S}}^{\prime }$ of the set $\mathcal{S}$ ofvertices in $\Delta$ and an automorphism group $G$strongly transitive and type preserving on $\Delta$. We prove thatthe algebra of $G$-invariant operators acting on the space offunctions on ${\mathcal{S}}^{\prime }$ is not commutative (contrarily to theclassical results) and we give its generators. We give also theprecise structure of some commutative subalgebras.