James Parkinson

Spherical harmonic analysis on affine buildings

Let be a locally finite regular affine building with root system R. There is a commutative algebra spanned by averaging operators Aλ, λ ∈ P+, acting on the space of all functions f:VP→, where VP is in most cases the set of all special vertices of , and P+ is a set of dominant coweights of R. This algebra is studied in [6] and [7] for Ãn buildings, and the general case is treated in [15].In this paper we show that all algebra homomorphisms h: 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 . We may regard as a subalgebra of the C*-algebra of bounded linear operators on ℓ2(VP), and we write for the closure of in this algebra. We study the Gelfand map , where M2=, and we compute M2 and the Plancherel measure of . We also compute the ℓ2-operator norms of the operators Aλ, λ ∈ P+, in terms of the Macdonald spherical functions.

A Local Limit Theorem for Random Walks on the Chambers of ˜ A 2 Buildings

In this paper we outline an approach for analysing random walks on the chambers of buildings. The types of walks that we consider are those which are well adapted to the structure of the building: Namely walks with transition probabilities p(c, d) depending only on the Weyl distance d(c, d). We carry through the computations for thick locally finite affine buildings of type A2 to prove a local limit theorem for these buildings. The technique centres around the representation theory of the associated Hecke algebra. This representation theory is particularly well developed for affine Hecke algebras, with elegant harmonic analysis developed by Opdam ([28], [29]). We give an introductory account of this theory in the second half of this paper.

AUTOMORPHISMS AND OPPOSITION IN TWIN BUILDINGS

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite. The main ingredient of the proof is a lemma that states that every duality of a thick finite projective plane admits an absolute point, i.e., a point mapped onto an incident line. Our results also hold for all finite irreducible spherical buildings of rank at least 3, and as a consequence we deduce that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

Asymptotic entropy of random walks on Fuchsian buildings and Kac-Moody groups

In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac–Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotropic random walk onto an apartment of the building, to prove existence of the asymptotic entropy for this retracted walk, and to ‘lift’ this in order to deduce the existence of the entropy for the random walk on the building.