We consider for an affine building of type Helgason's conjecture with respect to Laplace operators defined over different types of vertices. We prove that there are cases in which the conjecture fails, since there exist eigenfunctions which are not the Poisson transform of finitely additive measures at the maximal boundary of the building.
Let be thick affine building of type the Laplace operators of ; associated with a pair ; is the Poisson transform of a suitable finitely additive measure on the maximal boundary of ; by using only the combinatorial structure of .
In this paper we explicitly determine the Macdonald formula for spherical functions on any locally finite, regular and affine Bruhat-Tits building, by constructing the finite difference equations that must be satisfied and explaining how they arise, by only using the geometric properties of the building.
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