Normes invariantes et existence de filtrations admissibles.

In [Breuil et Schneider, J. reine angew. Math. 610: 149–180, 2007] is formulated a conjecture on the equivalence of the existence of invariant norms on certain locally algebraic representations of GL d+1( L) and the existence of certain de Rham representations of Gal(/ L), where L is a finite extension of ℚ p. In this paper, we prove the “easy” direction of the conjecture: the existence of invariant norms implies the existence of admissible filtrations. [ABSTRACT FROM AUTHOR]