Restriction of $p$-modular representations of $p$-adic groups to minimal parabolic subgroups

Abstract: Given a prime integer $p$, a non-archimedean local field $F$ of residual characteristic $p$ and a standard Borel subgroup $P$ of $GL_{2}(F)$, Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas 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 Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas 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 sheds a new light on Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas' 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.

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso