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

N. Abe (1) , G. Henniart (2, 3) , M.-F. Vignéras (4)

1 Hokkaido University [Sapporo, Japan]
2 UP11 UFR Sciences - Université Paris-Sud - Paris 11 - Faculté des Sciences
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay
4 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

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

Mots clés: Parabolic induction; pro-$p$ Iwahori Hecke algebra

Domaines: Mathématiques [math]

N. Abe, G. Henniart, M.-F. Vignéras. On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups. American Mathematical Society, 2018, 22 (5), pp.119 - 159. ⟨10.1090/ert/518⟩. ⟨hal-01919780⟩