(GLn+1(F), GLn(F)) is a Gelfand pair for any local field F

Avraham Aizenbud; Dmitry Gourevitch; Eitan Sayag

doi:10.1112/S0010437X08003746

(GL_n+1(F), GL_n(F)) is a Gelfand pair for any local field F

Avraham Aizenbud
, Dmitry Gourevitch
, Eitan Sayag

Research output: Contribution to journal › Article › peer-review

35 Scopus citations

Abstract

Let F be an arbitrary local field. Consider the standard embedding GL _n(F) → GL_n+1(F) and the two-sided action of GL _n(F)× GL_n(F) on GL_n+1(F). In this paper we show that any GL_n(F) × GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We show that this implies that the pair (GL_n+1(F), GL_n(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GL_n+1(F), Hom_GLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

Original language	English
Pages (from-to)	1504-1524
Number of pages	21
Journal	Compositio Mathematica
Volume	144
Issue number	6
DOIs	https://doi.org/10.1112/S0010437X08003746
State	Published - 1 Nov 2008
Externally published	Yes

Keywords

Invariant distribution
Multiplicity one

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1112/S0010437X08003746

Cite this

@article{70ddca52806c498a9982d711c4b7358d,

title = "(GLn+1(F), GLn(F)) is a Gelfand pair for any local field F",

abstract = "Let F be an arbitrary local field. Consider the standard embedding GL n(F) → GLn+1(F) and the two-sided action of GL n(F)× GLn(F) on GLn+1(F). In this paper we show that any GLn(F) × GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), HomGLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.",

keywords = "Invariant distribution, Multiplicity one",

author = "Avraham Aizenbud and Dmitry Gourevitch and Eitan Sayag",

year = "2008",

month = nov,

day = "1",

doi = "10.1112/S0010437X08003746",

language = "English",

volume = "144",

pages = "1504--1524",

journal = "Compositio Mathematica",

issn = "0010-437X",

publisher = "Cambridge University Press",

number = "6",

}

TY - JOUR

T1 - (GLn+1(F), GLn(F)) is a Gelfand pair for any local field F

AU - Aizenbud, Avraham

AU - Gourevitch, Dmitry

AU - Sayag, Eitan

PY - 2008/11/1

Y1 - 2008/11/1

N2 - Let F be an arbitrary local field. Consider the standard embedding GL n(F) → GLn+1(F) and the two-sided action of GL n(F)× GLn(F) on GLn+1(F). In this paper we show that any GLn(F) × GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), HomGLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

AB - Let F be an arbitrary local field. Consider the standard embedding GL n(F) → GLn+1(F) and the two-sided action of GL n(F)× GLn(F) on GLn+1(F). In this paper we show that any GLn(F) × GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), HomGLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

KW - Invariant distribution

KW - Multiplicity one

UR - https://www.scopus.com/pages/publications/56749177561

U2 - 10.1112/S0010437X08003746

DO - 10.1112/S0010437X08003746

M3 - Article

AN - SCOPUS:56749177561

SN - 0010-437X

VL - 144

SP - 1504

EP - 1524

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 6

ER -