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

Cartwright, Donald I. ; Kuhn, Gabriella

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 353-379

Résumé

Let $\mathfrak{X}$ be a homogeneous tree in which every vertex lies on $q+1$ edges, where $q\geq 2$. Let $\mathfrak{A}=Aut(\mathfrak{X})$ be the group of automorphisms of $\mathfrak{X}$, 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 $\pi$ 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 minimal subtree of $\pi$ is the smallest possible.