CHAPTER 0

I N T R O D U C T I O N

aba abc

cac

FIGURE la

Let G be a homogeneous (therefore infinite) tree of degree q + 1. Let each

edge of G be labelled with an integer between 1 and q + 1; let this be done in

such a way that of the q+1 edges containing any given vertex exactly one has

each label. Let G be the automorphism group of the labelled tree. A moment's

thought shows that G acts simply transitively on the vertex set of G. So, fixing

some vertex of G, we can identify the vertices of G with the elements of G. A

little further thought shows that G is (isomorphic to) the free product of q + 1

copies of Z2. Indeed, the q+1 automorphisms which map a fixed vertex to its

q+1 nearest neighbors constitute a set of q + 1 involutions which freely generate

G. See Figure la, where the involutions are denoted a, 6, and c. This paper is

devoted to harmonic analysis on G.1

Typically, harmonic analysis on a group G consists in finding a complete set of

irreducible representations of G, and then decomposing other representations in

1 The results of this paper constituted the Ph.D. dissertation of the second author, submitted

to Washington University, St. Louis, August 1985. Received by the editor July 17, 1986.

1