Buildings and Hecke algebras

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 S 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 and H the algebra B is isomorphic to H and the algebra A is isomorphic to the centre of H. 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. (c) 2005 Elsevier Inc. All rights reserved.