ar X iv : m at h / 06 04 05 6 v 1 [ m at h . R T ] 4 A pr 2 00 6 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 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 and H̃ , the algebra B is isomorphic to H and the algebra A is isomorphic to the centre of H̃ . 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. Introduction Let G = PGL(n + 1, F ) where F is a local field, and let K = PGL(n + 1,O), where O is the valuation ring of F . The space of bi-K-invariant compactly supported functions on G forms a commutative convolution algebra (see [18, Corollary 3.3.7] for example). Associated to G there is a building X (of type Ãn), and the above algebra is isomorphic to an algebra A of averaging operators defined on the space of all functions G/K → C. In [7] it was shown that these averaging operators may be defined in a natural way using only the geometric and combinatorial properties of X , hence removing the group G entirely from the discussion. For example, in the case n = 1, X is a homogeneous tree and A is the algebra generated by the operator A1, where for each vertex, (A1f)(x) is given by the average value of f over the neighbours of x. In [7], using this geometric approach, Cartwright showed that A is a commutative algebra, and that the algebra homomorphisms h : A → C can be expressed in terms of the classical Hall-Littlewood polynomials of [19, III, §2]. It was not assumed that X was constructed from a group G (although there always is such a group when n ≥ 3). Although not entirely realised in [7], as a consequence of our 2000 Mathematics Subject Classification. 20E42 (20C08 33D52 05E30 20N20).