. - /
, ab
^ \
terms of these irreducibles. However, this program can only be applied sensibly
to groups that are of type I, that is, groups for which the decomposition of a
representation into irreducibles is essentially unique. Our group G is not of type
I. So harmonic analysis on G does not follow the usual pattern. In this work,
we give a method for decomposing the regular representation, £2(G), into irre-
ducibles. In fact, this method gives many essentially inequivalent decompositions
of the regular representation.
Let pi, ,pg+i be positive real numbers with Y^j=\Pj = 1- Consider the
random walk on G given by the condition that the probability of jumping from
x to y is pj, whenever x and y are joined by an edge labelled j . The transition
matrix of this walk is denoted by R; Ryx = Pr(x y). As an operator, R is
bounded from £2(G) to £2(G). Note that R I is the natural discrete, anisotropic
Laplacian on the tree G.
Since the random walk described above is G-invariant, the operator R com-
mutes with the regular representation of G on £2(G). Thus, the generalized
eigenspaces, Ha, of R are preserved by the action of G. Consequently, G acts on
Ha according to a representation 7ra. This representation is, in fact, irreducible.
So when we write £2(G) as a direct integral of the generalized eigenspaces of R,
we have succeeded in decomposing the regular representation into irreducibles.
In general outline, and also in many of its details, this procedure for decompos-
ing the regular representation has an analogue in more traditional harmonic anal-
ysis. Let D be the disk { z C ; | 2 ; | l } , thought of as the hyperbolic plane. Let
dA be the hyperbolic area element on D, and let ALB be the Laplace-Beltrami
operator on D. Then 5L(2, R) acts on D through linear fractional transforma-
tions, and this action preserves dA and A ^ B - If we decompose L2(D,dA) into
generalized eigenspaces of A ^ B , then each generalized eigenspace is itself a rep-
resentation space for 5L(2,R). In fact, these representations form the spherical
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