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

Abstract

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.