Normes invariantes et existence de filtrations admissibles

Abstract

Let L be a finite extension of Q_p and d a positive integer. A conjecture, due to C. Breuil and P. Schneider, says that the existence of invariant norms on certain locally algebraic representations of GL_{d+1}(L) should be equivalent to the existence of certain (d+1)-dimensional de Rham representations of Gal(\bar{L}/L). We prove the easy direction of this conjecture: the existence of invariant norms implies the existence of admissible filtrations, by generalizing an idea of M.Emerton.