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

Abstract

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