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

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) \times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \times 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 $(\pi ,E)$ of (GL(n+1,F), $$dimHom_{GL(n,F)}(E,\cc) \leq 1.$$ For the proof in the archimedean case we develop several new tools to study invariant distributions on smooth manifolds.

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

References in corpus (3)

• Schwartz functions on Nash manifolds
• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F
• Multiplicity one Conjectures

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• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F
• Generalized Harish-Chandra descent and applications to Gelfand pairs
• A partial analog of integrability theorem for distributions on p-adic spaces and applications
• Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))
• An archimedean analog of Jacquet - Rallis theorem
• (GL(2n,C),SP(2n,C)) is a Gelfand Pair
• Multiplicity One Theorems and Invariant Distributions