Restricting cuspidal representations of the group of automorphisms of a homogeneous tree

Let cross X sign a homogeneous tree in which every vertex lies on q + 1 edges, where q ≥ 2. Let script a sign - Aut(cross X sign) be the group of automorphisms of cross X sign, and let H be the its subgroup PGL(2, F), where F is a local field whose residual field has order q. We consider the restriction to H of a continuous irreducible unitary representation π of script a sign. When π is spherical or special, it was well known that π remains irreducible, but we show that when π is cuspidal, the situation is much more complicated. We then study in detail what happens when the minimal subtree of π is the smallest possible