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

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

2007-09-09

arxiv.org/abs/0709.1273v4

Compositio Mathematica, Volume 144, pp 1504-1524 November 2008

Mathematics

Representation Theory

v3: Archimedean Localization principle excluded due to a gap in its proof. Another version of Localization principle can be fo

Scientific paper

10.1112/S0010437X08003746

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.

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