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

TL;DR

This paper proves that the pair (GL(n+1,F), GL(n,F)) forms a Gelfand pair over any local field F by analyzing invariant distributions and their symmetries, with new tools for the archimedean case.

Contribution

It establishes the Gelfand property for (GL(n+1,F), GL(n,F)) over all local fields, introducing new methods for studying invariant distributions.

Findings

Any GL(n,F) imes GL(n,F)-invariant distribution on GL(n+1,F) is symmetric under transposition.

The pair (GL(n+1,F), GL(n,F)) is a Gelfand pair, with multiplicity at most one.

New tools developed for the archimedean case to analyze invariant distributions.

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.