Représentations localement analytiques de ${\mathrm{GL}}_3\left({\mathbb{Q}}_p\right)$

Schraen, Benjamin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 43-145.

Nous construisons un complexe de représentations localement analytiques de ${\mathrm{GL}}_3\left({\mathbb{Q}}_p\right)$, associé à certaines représentations semi-stables de dimension $3$ du groupe de Galois absolu de ${\mathbb{Q}}_p$. Nous montrons ensuite que l’on peut retrouver le $(\varphi ,N)$-module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des $D\left({\mathrm{GL}}_3\left({\mathbb{Q}}_p\right)\right)$-modules, de ce complexe dans le complexe de de Rham de l’espace de Drinfel’d de dimension $2$. La preuve requiert le calcul de certains espaces de cohomologie localement analytiques de sous-groupes unipotents à coefficients dans des séries principales localement analytiques.

We construct a complex of locally analytic representations of ${\mathrm{GL}}_3\left({\mathbb{Q}}_p\right)$, which is associated to some semi-stable $3$-dimensional representations of the absolute Galois group of ${\mathbb{Q}}_p$. Then we show that we can retrieve the $(\varphi ,N)$-filtered module of the Galois representation in the space of morphisms, in the derived category of $D\left({\mathrm{GL}}_3\left({\mathbb{Q}}_p\right)\right)$-modules, of this complex in the de Rham-complex of the $2$-dimensional Drinfel’d’s space. For the proof, we compute some spaces of locally analytic cohomology of unipotent subgroups with coefficients in some locally analytic principal series.

Classification : 11F70, 11S20, 11S37, 11S80, 14G22, 22E50
Mot clés : correspondance de Langlands $p$-adique, espaces de Drinfel’d, représentations localement analytiques $p$-adiques
Keywords: $p$-adic Langlands correspondence, Drinfel’d’s spaces, $p$-adic locally analytic representations

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