Functorial properties of pro-$p$-Iwahori cohomology

Abstract

Suppose $F$ is a finite extension of ${\mathbb{Q}}_p$, $G$ is the group of $F$-points of a connected reductive $F$-group, and $I_1$ is a pro-$p$-Iwahori subgroup of $G$. We construct two spectral sequences relating derived functors on mod-$p$ representations of $G$ to the analogous functors on Hecke modules coming from pro-$p$-Iwahori cohomology. More specifically: (1) using results of Ollivier--Vignéras, we provide a link between the right adjoint of parabolic induction on pro-$p$-Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a "Poincaré duality spectral sequence" relating duality on pro-$p$-Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules ${\text{H}}^i(I_1,\pi )$.

35 pages. Minor changes following referee report. To appear in JLMS

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• Parabolic induction and extensions
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Cited by in corpus (1)

• Derived right adjoints of parabolic induction: an example