Let G be a locally compact
group and π : G →𝒰(ℋ) a unitary representation of G. A commutative subalgebra of
ℬℋ is called π-inductive when it is stable through conjugation by every operator in
the range of π. This concept generalizes Mackey’s definition of a system of
imprimitivity for π; it is expected that studying inductive algebras will lead to
progress in the classification of realizations of representations on function spaces. In
this paper we take as G the automorphism group of a locally finite homogeneous tree;
we consider the principal spherical representations of G, which act on a Hilbert space
of functions on the boundary of the tree, and classify the maximal inductive algebras
of such representations. We prove that, in most cases, there exist exactly two such
algebras.
