Inductive algebras for trees

Let G be a locally compact group and pi : G --> U(H) a unitary representation of G. A commutative subalgebra of B(H) is called pi-inductive when it is stable through conjugation by every operator in the range of pi. This concept generalizes Mackey's definition of a system of imprimitivity for pi; 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.