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In this paper we establish a strong connection between buildings and Hecke algebras by studying two algebras of averaging operators on buildings. To each locally finite regular building we associate a natural algebra B of chamber set averaging operators, and when the building is affine we also define an algebra A of vertex set averaging operators. We show that for appropriately parametrised Hecke algebras H andH , the algebra B is isomorphic to H and the algebra A is isomorphic to the centre ofH. On the one hand these results give a thorough understanding of the algebras A and B. On the other hand they give a nice geometric and combinatorial understanding of Hecke algebras, and in particular of the Macdonald spherical functions and the centre of affine Hecke algebras. Our results also produce interesting examples of association schemes and polynomial hypergroups. In later work we use the results here to study random walks on affine buildings.

The Littelmann path model gives a realization of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent and Littelmann [S. Gaussent, P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (1) (2005) 35-88] and others [A. Braverman, D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (3) (2001) 561-575; S. Gaussent, G. Rousseau, Kac-Moody groups, hovels and Littelmann's paths, preprint, arXiv: math.GR/0703639, 2007] has demonstrated a connection between this model and the geometry of the loop Grassmanian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the cells in generalized Mirković-Vilonen intersections.

Let θ be an automorphism of a thick irreducible spherical building ∆ of rank at least 3 with no Fano plane residues. We prove that if there exist both type J 1 and J 2 simplices of ∆ mapped onto opposite simplices by θ, then there exists a type J 1 ∪ J 2 simplex of ∆ mapped onto an opposite simplex by θ. This property is called cappedness. We give applications of cappedness to opposition diagrams, domesticity, and the calculation of displacement in spherical buildings. In a companion piece to this paper we study the thick irreducible spherical buildings containing Fano plane residues. In these buildings automorphisms are not necessarily capped.

Abstract. 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] forÃn 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.