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

Cartwright, Donald I.; Kuhn, Gabriella

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 353-379 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

Let $X$ be a homogeneous tree in which every vertex lies on $q + 1$ edges, where $q \geq 2$ . Let $A = A u t (X)$ be the group of automorphisms of $X$ , and let $H$ be the its subgroup $P G L (2, F)$ , 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 $A$ . 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.

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

Publié le : 2003-06-01

MR 1988210 | Zbl 1177.43008

@article{BUMI_2003_8_6B_2_353_0,
     author = {Donald I. Cartwright and Gabriella Kuhn},
     title = {Restricting cuspidal representations of the group of automorphisms of a homogeneous tree},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {6-A},
     year = {2003},
     pages = {353-379},
     zbl = {1177.43008},
     mrnumber = {1988210},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2003_8_6B_2_353_0}
}

Cartwright, Donald I.; Kuhn, Gabriella. Restricting cuspidal representations of the group of automorphisms of a homogeneous tree. Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003) pp. 353-379. http://gdmltest.u-ga.fr/item/BUMI_2003_8_6B_2_353_0/

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