Socle localement analytique I

Breuil, Christophe

Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 633-685.

Soit $L$ une extension finie de ${\mathbf{Q}}_p$ et $n$ un entier $>0$. À toute filtration de Hodge de poids de Hodge-Tate distincts sur une représentation de rang $n$ suffisamment générique du groupe de Weil-Deligne de $L$, on associe une représentation localement ${\mathbf{Q}}_p$-analytique semi-simple de longueur finie de ${\mathrm{GL}}_n\left(L\right)$. On montre plusieurs propriétés de cette représentation. Par exemple, lorsqu’elle possède un réseau stable par ${\mathrm{GL}}_n\left(L\right)$, alors la filtration de départ est faiblement admissible.

Let $L$ be a finite extension of ${\mathbf{Q}}_p$ and $n$ a positive integer. To each Hodge filtration with distinct Hodge-Tate weights on an $n$-dimensional sufficiently generic representation of the Weil-Deligne group of $L$, we associate a semi-simple finite length locally ${\mathbf{Q}}_p$-analytic representation of ${\mathrm{GL}}_n\left(L\right)$. We show several properties of this representation of ${\mathrm{GL}}_n\left(L\right)$. For instance, if it has an invariant lattice, then the starting Hodge filtration is weakly admissible.

Classification : 11S23, 22E35, 22E50
Mot clés : Représentation localement analytique, filtration de Hodge, socle
Keywords: Locally analytic representation, Hodge filtration, socle

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