Macdonald formula for spherical functions on affine buildings (Q661134)

Let \(X\) be a locally finite affine building. To \(X\) is associated an algebra of averaging operators, called its Hecke algebra. If \(X\) is associated to a semi-simple \(p\)-adic Lie group \(G\), with maximal compact subgroup \(K\), this algebra is the algebra \(\mathcal L\) of continuous bi-\(K\)-invariant functions on \(G\) with compact support. Spherical functions on \(X\) are (roughly) defined as eigenfunctions of this algebra. The Macdonald formula, proved originally by Macdonald for semi-simple \(p\)-adic groups [\textit{I. G. Macdonal}, Publ. Ramanujan Inst. 2, 79 p. (1971; Zbl 0302.43018)], gives a formula for any spherical function in terms of a character on the coweight lattice. The object of this paper is to prove this formula in a purely geometric way, without any reference to \(p\)-adic groups. Thus, it also applies to the buildings (of rank \(2\), by Tits' classification) which are not associated to any \(p\)-adic group. An important tool in the proof of this formula is the boundary of the building, and the expression of spherical functions as integrations of a Poisson kernel on the boundary. The same results were independently proved, with a different method, by \textit{J. Parkinson} [Math. Z. 253, No. 3, 571--606 (2006; Zbl 1171.43009)].