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

Part of: Linear algebraic groups and related topics Lie groups Distributions, generalized functions, distribution spaces

Published online by Cambridge University Press:  01 November 2008

Avraham Aizenbud ,
Dmitry Gourevitch  and
Eitan Sayag
Avraham Aizenbud
Affiliation:
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel (email: aizenr@yahoo.com)
Dmitry Gourevitch
Affiliation:
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel (email: guredim@yahoo.com)
Eitan Sayag
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (email: eitan.sayag@gmail.com)
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Abstract

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Let F be an arbitrary local field. Consider the standard embedding and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), . For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

Keywords

multiplicity oneinvariant distribution

MSC classification

Secondary: 22E45: Representations of Lie and linear algebraic groups over real fields 20G05: Representation theory 20G25: Linear algebraic groups over local fields and their integers 46F99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 6 , November 2008 , pp. 1504 - 1524
Copyright
Copyright © Foundation Compositio Mathematica 2008