Spherical harmonic analysis on affine buildings

Abstract

Let be a locally finite regular affine building with root system R. There is a commutative algebra spanned by averaging operators A λ , λ ∈ P+, acting on the space of all functions f:V P → , where V P is in most cases the set of all special vertices of , and P+ is a set of dominant coweights of R. This algebra is studied in [6] and [7] for à n buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h: 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 . We may regard as a subalgebra of the C*-algebra of bounded linear operators on ℓ2(V P ), and we write for the closure of in this algebra. We study the Gelfand map , where M2= , and we compute M2 and the Plancherel measure of . We also compute the ℓ2-operator norms of the operators A λ , λ ∈ P+, in terms of the Macdonald spherical functions.