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

Schraen, Benjamin

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

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.

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.

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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