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

Abstract

We consider a building $\Delta$ of type ${\tilde{B}}_n(n\ge 3)$, different subsets ${\mathcal{S}}^{\prime }$ of the set $\mathcal{S}$ of vertices in $\Delta$ and an automorphism group $G$ strongly transitive and type preserving on $\Delta$. We prove that the algebra of $G$-invariant operators acting on the space of functions on ${\mathcal{S}}^{\prime }$ is not commutative (contrarily to the classical results) and we give its generators. We give also the precise structure of some commutative subalgebras.