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

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 ${\mathrm{I}\mathrm{n}\mathrm{d}}_P^G\tau$ and we show that ${\mathrm{I}\mathrm{n}\mathrm{d}}_P^G\chi \tau$ is irreducible for a general unramified character $\chi$ of $M$. In the reverse direction, we compute the image by the two adjoints of ${\mathrm{I}\mathrm{n}\mathrm{d}}_P^G$ of an irreducible admissible representation $\pi$ of $G$. On the way, we prove that the right adjoint of ${\mathrm{I}\mathrm{n}\mathrm{d}}_P^G$ respects admissibility, hence coincides with Emerton’s ordinary part functor ${\mathrm{O}\mathrm{r}\mathrm{d}}_{\bar{P}}^G$ on admissible representations.