In this paper we define generalised spheres in buildings using the simplicial
structure and Weyl distance in the building, and we derive an explicit formula
for the cardinality of these spheres. We prove a generalised notion of distance
regularity in buildings, and develop a combinatorial formula for the
cardinalities of intersections of generalised spheres. Motivated by the
classical study of algebras associated to distance regular graphs we
investigate the algebras and modules of Hecke operators arising from our
generalised distance regularity, and prove isomorphisms between these algebras
and more well known parabolic Hecke algebras. We conclude with applications of
our main results to non-negativity of structure constants in parabolic Hecke
algebras, commutativity of algebras of Hecke operators, double coset
combinatorics in groups with BN-pairs, and random walks on the simplices of
buildings.Comment: J. Algebra, to appea
