On Laplace operators on buildings

A rank $l$ Riemannian symmetric space $Z$ is a homogeneous space under the group $G$ of real points of a connected reductive group. The points in $Z$ correspond to the Cartan involutions on $G$, the maximal flat subspaces in $Z$ correspond to the maximal split tori for $G$, the distinguished boundary of $Z$ to the set of minimal parabolic subgroups. The Cartan decomposition realizes a map from $G\setminus(Z\times Z)$ to a real cone of dimension $l$, and this defines spheres on the space $Z$. The ring of invariant differential operators on $Z$ is a rank $l$ polynomial algebra. The joint eigenfuncions of this algebra are also the eigenfunctions for the average operators on spheres. As conjectured by Helgason, they can be expressed as suitable Poisson integral over the boundary.
With K. F. Lai from Sydney, we develop a strategy to define the similar objects on a rank $l$ Bruhat-Tits building $Z$, which is not always associated to a reductive group : spheres, horocycles, apartments, average operators, distinguished bounbdary, …and then to obtain a Poisson realization on the distinguished boundary for the joint eigenfunctions on $Z$. The case of type $A$ building is now completed.