On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups

Let $F$ be locally compact field with residue characteristic $p$, and $\mathbf{G}$ a connected reductive $F$-group. Let $\mathcal{U}$ be a pro-$p$ Iwahori subgroup of $G=\mathbf{G}(F)$. Fix a commutative ring $R$. If $\pi$ is a smooth $R[G]$-representation, the space of invariants ${\pi }^{\mathcal{U}}$ is a right module over the Hecke algebra $\mathcal{H}$ of $\mathcal{U}$ in $G$. Let $P$ be a parabolic subgroup of $G$ with a Levi decomposition $P=MN$ adapted to $\mathcal{U}$. We complement previous investigation of Ollivier-Vign'eras on the relation between taking $\mathcal{U}$-invariants and various functor like ${\mathrm{I}\mathrm{n}\mathrm{d}}_P^G$ and right and left adjoints. More precisely the authors’ previous work with Herzig introduce representations $I_G(P,\sigma ,Q)$ where $\sigma$ is a smooth representation of $M$ extending, trivially on $N$, to a larger parabolic subgroup $P(\sigma )$, and $Q$ is a parabolic subgroup between $P$ and $P(\sigma )$. Here we relate $I_G(P,\sigma ,Q{)}^{\mathcal{U}}$ to an analogously defined $\mathcal{H}$-module $I_{\mathcal{H}}(P,{\sigma }^{{\mathcal{U}}_M},Q)$, where $$\mathcal{U}$_M = $\mathcal{U}$\cap M$and$\sigma^{$\mathcal{U}$_M}$ is seen as a module over the Hecke algebra ${\mathcal{H}}_M$ of ${\mathcal{U}}_M$ in $M$. In the reverse direction, if $\mathcal{V}$ is a right ${\mathcal{H}}_M$-module, we relate $I_{\mathcal{H}}(P,\mathcal{V},Q)\otimes {\text{c-Ind}}_{\mathcal{U}}^G\mathbf{1}$ to $I_G(P,\mathcal{V}{\otimes }_{{\mathcal{H}}_M}{\text{c-Ind}}_{{\mathcal{U}}_M}^M\mathbb{1},Q)$. As an application we prove that if $R$ is an algebraically closed field of characteristic $p$, and $\pi$ is an irreducible admissible representation of $G$, then the contragredient of $\pi$ is $0$ unless $\pi$ has finite dimension.