Normes invariantes et existence de filtrations admissibles

TL;DR

This paper proves that the existence of invariant norms on certain representations implies the existence of admissible filtrations, supporting a conjecture linking representation theory and Galois representations over p-adic fields.

Contribution

It establishes one direction of a conjecture by Breuil and Schneider, connecting invariant norms to admissible filtrations in p-adic representation theory.

Findings

Invariant norms imply admissible filtrations.

Generalization of Emerton's idea to the p-adic setting.

Supports the conjectural link between representations and Galois theory.

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.