Eigenspaces of operator algebras on the fundamental apartment of an affine building

We consider, on the fundamental apartment of any affine building, the operator algebras obtained as retraction of the vertex set Hecke algebra of the building, with respect to a boundary point and a chamber respectively. We characterize the dimension of the eigenspaces of both these operator algebras. Then we apply this characterization to investigate on buildings the classical Helgason's result.