  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 $\mathrm{Ind}_P^G$ 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
$I_\mathcal{H}(P,\mathcal{V},Q)\otimes \textrm{c-Ind}_\mathcal{U}^G\mathbf{1}$
to
$I_G(P,\mathcal{V}\otimes_{\mathcal{H}_M}\textrm{c-Ind}_{\mathcal{U}_M}^M\mathbb{1},Q)$.
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.
