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
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