ar X iv : m at h / 06 04 05 8 v 1 [ m at h . FA ] 4 A pr 2 00 6 SPHERICAL HARMONIC ANALYSIS ON AFFINE BUILDINGS

Let X be a locally finite regular affine building with root system R. There is a commutative algebra A spanned by averaging operators A λ , λ ∈ P + , acting on the space of all functions f : V P → C, where V P is in most cases the set of all special vertices of X , and P + is a set of dominant coweights of R. This algebra is studied in [6] and [7] for˜An buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h : A → C may be expressed in terms of the Macdonald spherical functions. We also provide a second formula for these homomorphisms in terms of an integral over the boundary of X. We may regard A as a subalgebra of the C *-algebra of bounded linear operators on ℓ 2 (V P), and we write A 2 for the closure of A in this algebra. We study the Gelfand map A 2 → C (M 2), where M 2 = Hom(A 2 , C), and we compute M 2 and the Plancherel measure of A 2. We also compute the ℓ 2-operator norms of the operators A λ , λ ∈ P + , in terms of the Macdonald spherical functions.