This paper is concerned with the action of a special formally real Jordan algebra U on an Euclidean space E, with the decomposition of E under this action and with an application of this decomposition to the study of Bessel functions on the self-adjoint homogeneous cone associated to U.

Let $\mathfrak{X}$ be a homogeneous tree in which every vertex lies on $q+1$ edges, where $q\ge 2$. Let $\mathfrak{A}=Aut\left(\mathfrak{X}\right)$ be the group of automorphisms of $\mathfrak{X}$, and let $H$ be the its subgroup $PGL\left(2,F\right)$, where $F$ is a local field whose residual field has order $q$. We consider the restriction to $H$ of a continuous irreducible unitary representation $\pi$ of $\mathfrak{A}$. When $\pi$ is spherical or special, it was well known that $\pi$ remains irreducible, but we show that when $\pi$ is cuspidal, the situation is much more complicated. We then study in detail what happens when the...