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

• Published in: Representation theory Vol. 22; no. 5; pp. 119 - 159
• Main Authors: N. Abe, G. Henniart, M.-F. Vignéras
• Format: Journal Article
• Language: English
• Published: 15.10.2018
• Subjects: pro-$p$ Iwahori Hecke algebra; Parabolic induction
• ISSN: 1088-4165
• DOI: 10.1090/ert/518

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.