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

Let $F$ be a local field with finite residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf G$ be a connected reductive $F$-group. With Abe, Henniart, Herzig, we classified irreducible admissible $C$-representations of $G=\mathbf G(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. 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 $\mathrm{Ind}_P^G \tau$. In the reverse direction, we compute the image by the two adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation $\pi$ of $G$. We prove that the smooth dual of $\pi$ is $0$ unless $\pi$ is finite dimensional. If $\mathcal U$ is a pro-$p$ Iwahori subgroup of $G$, we determine the space of $\mathcal U$-invariants $\pi^{\mathcal U}$ as a module over the Hecke algebra of $\mathcal U$ in $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G$ respects admissibility, hence coincides with Emerton's ordinary part functor on admissible representations. This is common work with Abe and Henniart.