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.

Abstract (VO)

Let $\mathfrak{X}$ be a homogeneous tree in which every vertex lies on $q+1$ edges, where $q\ge 2$. 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.

Riassunto

Sia $\mathfrak{X}$ un albero omogeneo dove a ogni vertice si incontrano $q+1$$(q\ge 2)$ spigoli. Sia $\mathfrak{A}=Aut(\mathfrak{X})$ il gruppo di automorfismi di $\mathfrak{X}$ e $H$ un sottogruppo chiuso isomorfo a $PGL(2,F)$ ($F$ campo locale il cui campo residuo ha ordine $q$). Sia $\pi$ una rappresentazione continua unitaria e irriducibile di $\mathfrak{A}$ e si consideri ${\pi }_H$, la sua restrizione ad $H$. È noto che se $\pi$ è una rappresentazione sferica o speciale ${\pi }_H$ rimane irriducibile. In questo lavoro si mostra che quando $\pi$ è cuspidale la situazione è molto più complessa. Si studia in dettagli o il caso in cui il sotto albero minimale associato a $\pi$ sia il più piccolo possibile, ottenendo una esplicita decomposizione di ${\pi }_H$.

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