Séminaire de géométrie arithmétique
Given a prime integer p, a non-archimedean local field F of residual characteristic p and a standard Borel subgroup P of ${GL}_{2} (F)$, Paškūnas proved that the restriction to P of (irreducible) smooth representations of ${GL}_{2} (F)$ over $\overline{\mathbb F}_p$ encodes a lot of information about the full representation of ${GL}_{2} (F)$ and that it leads to useful statement about p-adic representations of ${GL}_{2} (F)$. Nevertheless, the methods used at that time by Paškūnas heavily relied on the understanding of the action of certain spherical Hecke operator and on some combinatorics specific to the ${GL}_{2} (F)$ case. This method can be transposed case by case to for some other quasi-split groups of rank 1, but this is not very satisfying as such.
This talk will report on a joint work with J. Hauseux. Using Emerton's ordinary parts functor, we get a more uniform context which shed a new light on Paškūnas' results and allows us to generalize very naturally these results for arbitrary rank 1 groups. In particular, we prove that for such groups, the restriction of supersingular representations to a minimal parabolic subgroup is always irreducible.
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Christophe Soulé et Jeanine Van Order
