Sur la fid\'elit\'e de certaines repr\'esentations de GL_2(F) sous une alg\`ebre d'Iwasawa

TL;DR

This paper proves that certain ideals in the Iwasawa algebra are open and demonstrates the faithfulness of the action of upper unipotent matrices on irreducible admissible smooth representations of GL2 over a finite extension of Qp.

Contribution

It establishes the openness of O_F*-stable ideals in the Iwasawa algebra and confirms the faithful action of E[[U]] on irreducible admissible smooth representations of GL2(F).

Findings

Non-zero O_F*-stable ideals in E[[O_F]] are open.

The action of E[[U]] on certain representations is faithful.

Results hold for finite extensions of Qp.

Abstract

Let F be a finite extension of Qp, O_F its ring of integers and E a finite extension of Fp. The natural action of the unit group O_F* on O_F extends in a continuous action on the Iwasawa algebra E[[O_F]]. In this work, we show that non zero ideals of E[[O_F]] which are stable under O_F* are open. As a consequence, we deduce the fidelity of the action of E[[U]], with U the subgroup of upper unipotent matrices in GL2(O_F) on an irreducible admissible smooth E-representation of GL2(F). ----- Soit F une extension finie de Qp, d'anneau des entiers O_F et E une extension finie de Fp. L'action naturelle du groupes des unit\'es O_F* sur O_F se prolonge alors en une action continue sur l'alg\`ebre d'Iwasawa E[[O_F]]. Dans ce travail, on d\'emontre que les id\'eaux non nuls de E[[O_F]] stables par O_F* sont ouverts. En particulier, on en d\'eduit la fid\'elit\'e de l'action de l'alg\`ebre d'Iwasawa des matrices unipotentes sup\'erieures de GL2(O_F) sur une repr\'esentation lisse irr\'eductible admissible de GL2(F).