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Comptes Rendus Mathématique
Volume 353, n° 7
pages 579-582 (juillet 2015)
Doi : 10.1016/j.crma.2015.04.002
Received : 24 December 2014 ;  accepted : 3 April 2015
A note on the Kirillov model for representations of  
Une note sur le modèle de Kirillov des représentations de  
 

Alexander Kemarsky
 Technion, Mathematics, Department of Mathematics, Haifa, Israel 

Abstract

Let   and   be an additive character. Let U be the subgroup of upper triangular unipotent matrices in G . Denote by θ the character   given by
θ(u):=ψ(u1,2+u2,3+…+un−1,n). Let P be the mirabolic subgroup of G consisting of all matrices in G with the last row equal to  . We prove that if π is an irreducible generic representation of   and   is its Whittaker model, then the space   contains the space of infinitely differentiable functions   that satisfy   for all   and   and that have a compact support modulo U . A similar result was proven for  , where F is a p -adic field by Gelfand and Kazhdan (1975) [[1]] and for   by Jacquet (2010) [[2]].

The full text of this article is available in PDF format.
Résumé

Soit   et   un caractère additif non trivial. Soit U le sous-groupe des matrices triangulaires supérieures unipotentes de G . Notons   le caractère donné par
θ(u):=ψ(u1,2+u2,3+⋯+un−1,n). Soit P le sous-groupe mirabolique constitué des matrices de G dont la dernière ligne est  . Nous montrons que, si π est une représentation irréductible générique de G et si   est son modèle de Whittaker, alors l'espace   contient l'espace des fonctions   infiniment différentiables, qui satisfont   pour tout   et   et qui ont un support compact modulo U . Un résultat similaire a été établi pour  , où F est un corps p -adique, par Gelfand et Kazhdan (1975) [[1]] et pour   par Jacquet (2010) [[2]].

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