Spherical harmonic analysis on affine buildings

To each regular affine building there is naturally associated a commutative algebra A spanned by vertex set averaging operators. In this paper we study the algebra homomorphisms from A into the complex numbers. In particular, we provide two explicit formulae for these homomorphisms; one in terms of the Macdonald spherical functions, and another as an integral over the boundary of the building. We also determine the associated Plancherel measure, and we compute the l^2-operator norms of each basis element of A.