Buildings and Hecke algebras

Abstract

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.

Spherical harmonic analysis on affine buildings

“…We have I P = {i ∈ I | m i = 1}, which shows that 0 ∈ I P for all root systems, and that I P = {0} if R is non-reduced [15,Lemma 4.3]. This also shows that in the non-reduced case there are 'special' vertices which are not 'good' vertices.…”

“…The contents of this paper (and [15]) lead us to the conclusion that, rather than playing a relatively minor role, the building theoretic elements alone determine the nature of the algebra L (G, K) and the zonal spherical functions.…”

“…In [15] we showed that under the weak hypothesis of regularity, to each affine building X of irreducible type there is naturally associated a commutative algebra A spanned by vertex set averaging operators A λ , λ ∈ P + , acting on the space of all functions f : V P → C. Here P + is the set of dominant coweights of an appropriate root system R, and V P is in most cases the set of all special vertices of X .…”

“…See [8, Propositions 2.4 and 2.5] for details in the R = A 2 case. In [15,Theorem 6.16] we showed that A ∼ = C[P ]…”

“…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 .…”