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arXiv · 0709.1273

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

Avraham Aizenbud·Dmitry Gourevitch·Eitan Sayag
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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 $(π,E)$ 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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Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag. 2009-05-17. (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F. https://doi.org/10.1112/s0010437x08003746

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