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.
Categories: math.NT math.RT