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

Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

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.

Original languageEnglish
Pages (from-to)1504-1524
Number of pages21
JournalCompositio Mathematica
Volume144
Issue number6
DOIs
StatePublished - 1 Nov 2008

Keywords

  • Mathematics - Representation Theory
  • 22E,22E45,20G05,20G25,46F99

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