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arXiv · 1105.5786

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

Hu Yongquan·Morra Stéfano·Schraen Benjamin
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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).

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Hu Yongquan, Morra Stéfano, Schraen Benjamin. 2011-05-29. Sur la fid\'elit\'e de certaines repr\'esentations de GL_2(F) sous une alg\`ebre d'Iwasawa. https://doi.org/10.4171/rsmup%2F131-4

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