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

TL;DR

This paper explores the properties of modulo p representations of reductive p-adic groups, focusing on induction, irreducibility, and adjoint functors, extending the understanding of their functorial behavior and subrepresentation structures.

Contribution

It characterizes subrepresentation lattices of induced representations and establishes the behavior of induction and adjoint functors in the modulo p setting for reductive p-adic groups.

Findings

Induction of unramified twists remains irreducible.

The right adjoint of induction coincides with Emerton's ordinary parts functor.

The lattice of subrepresentations of induced representations is explicitly determined.

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.