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

Abstract:

Let $F$ be a locally compact field with residue characteristic $p$, and let $\mathbf {G}$ be 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 a previous investigation of Ollivier-Vignéras on the relation between taking $\mathcal {U}$-invariants and various functor like $\operatorname {Ind}_P^G$ and right and left adjoints. More precisely the authors’ previous work with Herzig introduced 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 $I_\mathcal {H}(P,\mathcal {V},Q)\otimes \operatorname {c-Ind}_\mathcal {U}^G\mathbf {1}$ to $I_G(P,\mathcal {V}\otimes _{\mathcal {H}_M}\operatorname {c-Ind}_{\mathcal {U}_M}^M\mathbf {1},Q)$. 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.