(O(V+F), O(V)) is a Gelfand pair for any quadratic space V over a local field F

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dc.creatorAizenbud, Avraham
dc.creatorGourevitch, Dmitry
dc.creatorSayag, Eitan
dc.date2007-11-09
dc.date2009-05-17
dc.date.accessioned2026-07-07T13:15:14Z
dc.date.available2026-07-07T13:15:14Z
dc.descriptionLet V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $W=V \oplus Fe$ with the form Q extending q with Q(e)=1. Consider the standard embedding of O(V) into O(W) and the two-sided action of $O(V)\times O(V)$ on $O(W)$. In this note we show that any $O(V)\times O(V)$-invariant distribution on O(W) is invariant with respect to transposition. This result was earlier proven in a bit different form in [vD] for F=R, in [AvD] for F=C and in [BvD] for p-adic fields. Here we give a different proof. Using results from [AGS], we show that this result on invariant distributions implies that the pair (O(V),O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Frechet representation E of O(W) we have $dim Hom_{O(V)}(E,C) \leq 1.$ A stronger result for p-adic fields is obtained in [AGRS].
dc.description6 pages. v4: Formulation of Localization principle changed. v5: a minor change
dc.identifierhttps://arxiv.org/abs/0711.1471
dc.identifierhttp://arxiv.org/abs/0711.1471
dc.identifierMathematische Zeitschrift. Volume 261, Number 2 Feb 2009
dc.identifierdoi:10.1007/s00209-008-0318-5
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/230414
dc.subjectRepresentation Theory
dc.subject22E,22E45,20G05,20G25,46F99
dc.title(O(V+F), O(V)) is a Gelfand pair for any quadratic space V over a local field F
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