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

2007/09/30 by Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag · 1 citation
Mathematics · #math.RT #msc:22E #msc:22E45 #msc:20G05 #msc:20G25 #msc:46F99

paper · pdf · doi:10.1112/s0010437x08003746

published as Compositio Mathematica, Volume 144, pp 1504-1524 November 2008 · v3: Archimedean Localization principle excluded due to a gap in its proof. Another version of Localization principle can be found in arXiv:0803.3395v2 [RT]. v4: an inaccuracy with Bruhat filtration fixed. See Theorem 4.2.1 and Appendix B

arxiv created 2009/05/17 · arxiv updated 2009/12/01

Abstract

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into 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), dimHom_GL(n,F)(E,\cc) ≤ 1. For the proof in the archimedean case we develop several new tools to study invariant distributions on smooth manifolds.

Citations

• Generalized Harish-Chandra descent and applications to Gelfand pairs

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