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

Donald I. Cartwright; Gabriella Kuhn

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 353-379
  • ISSN: 0392-4041

Abstract

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Let X be a homogeneous tree in which every vertex lies on q + 1 edges, where q ≥ 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.

How to cite

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Cartwright, Donald I., and Kuhn, Gabriella. "Restricting cuspidal representations of the group of automorphisms of a homogeneous tree." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 353-379. <http://eudml.org/doc/195672>.

@article{Cartwright2003,
abstract = {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.},
author = {Cartwright, Donald I., Kuhn, Gabriella},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {353-379},
publisher = {Unione Matematica Italiana},
title = {Restricting cuspidal representations of the group of automorphisms of a homogeneous tree},
url = {http://eudml.org/doc/195672},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Cartwright, Donald I.
AU - Kuhn, Gabriella
TI - Restricting cuspidal representations of the group of automorphisms of a homogeneous tree
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 353
EP - 379
AB - 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.
LA - eng
UR - http://eudml.org/doc/195672
ER -

References

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