Harmonic analysis, Hecke algebra and cohomology on groups of trees and buildings

Abstract

The PhD project consists of two parts. The first part is about finite trees, realizations and random walks. The second part is about the Hecke algebras of infinite trees and buildings and the cohomology groups. We note that some examples of finite trees can be generalized to the infinite cases. The words of finite length in the example of ultrametric rooted trees with finite depth can be extended to doubly infinite chains in the infinite homogeneous trees thus defining a Banach algebra. As the background of the project, we study the topics of finite phylogenetic trees by understanding the combinatorial and geometrical structure of rooted and unrooted discrete BHV tree space of n taxa. Certain types of random walks on the space of trees can be used to model the evolution process. As a method to improve the computation of such random walks, we realize some tree spaces into polytopes in Euclidean space where the vertices, edges and faces indicate trees of different degenerate levels. In particular, we study the links between the permuto-associahedra and the BHV tree-space. One specific realization is called the secondary polytope, which is used to construct the associahedra, and we will generalize this construction into more complicated examples and compare with the BME polytopes of the BHV trees. In order to study the random walks on tree space, we apply several classic methods such as the eigensolutions of Markov chains, Gelfand pairs and spherical functions to decompose the functions on tree space. We present some classic examples where these methods solve the random walk explicitly. We consider biinvariant subalgebras of group algebra which are commutative under convolution. These arise from Gelfand pairs where spherical functions can be used to produce the eigenvectors of the transition matrix of the random walk. We note that an example of the q-homogeneous rooted tree of a finite depth is a good link to generalize the study from finite to infinite cases where the space is still discrete. The first example in the second part of the thesis is the infinite homogeneous trees and we study the invariant subalgebra under the ` 1 norm. The space can be discretized to Z+ and we show that it is isomorphic to a Hecke algebra with single generator, the Hecke operator which corresponds to the random walk generator. It is natural to consider some key properties of the algebra, i.e. the spherical functions, character space, derivations and b.a.i.. The main example we study is the Gelfand pair given by projective general linear groups over p-adic numbers and the subgroup corresponding to the the p-adic integers, where the example of the smallest dimension corresponds to infinite homogeneous trees and examples of higher dimensions correspond to the Bruhat-Tits buildings of type A˜. We claim that the Hecke algebras of these Gelfand pairs are isomorphic to the invariant subalgebras of functions on the A˜ lattice subject to weight conditions determined by p. Based on the isomorphic algebras on the type A˜ lattices, we consider the examples of types A˜ and B˜, with and without the invariance conditions under the Weyl group action on the lattices. We show that the above examples are all finitely generated and the number of generators in each case are equal to the dimension of the lattice in the Euclidean space. We then compute cohomology groups of the algebras of functions on the weighted lattices. We build up from the methods introduced in the examples of those similar to Z+ and Z k + . The general idea is to calculate the approximate formulae from the precise ones in Z+ and Z k + and iterate the process with an induction by reducing the degree of the leading terms. We also expect this method can be generalized to the Hecke algebras of other Gelfand pairs with corresponding weighted lattices.