# Modulo $p$ representations of reductive $p$-adic groups: functorial properties

Abstract : Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $\tau$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$ and we show that $\mathrm{Ind}_P^G \chi \tau$ is irreducible for a general unramified character $\chi$ of $M$. In the reverse direction, we compute the image by the two adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation $\pi$ of $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G$ respects admissibility, hence coincides with Emerton's ordinary part functor $\mathrm{Ord}_{\overline{P}}^G$ on admissible representations.
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https://hal.sorbonne-universite.fr/hal-01919518
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Submitted on : Monday, November 12, 2018 - 3:12:45 PM
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• HAL Id : hal-01919518, version 1
• ARXIV : 1703.05599

### Citation

Noriyuki Abe, Guy Henniart, Marie-France Vignéras. Modulo $p$ representations of reductive $p$-adic groups: functorial properties. 2018. ⟨hal-01919518⟩

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