On the Right Derived Functors of Ordinary Parts

TL;DR

This paper proves a variant of Emerton's conjecture on the right derived functors of the ordinary parts functor in mod p representations of p-adic groups, providing explicit descriptions and applications to Bernstein's Second Adjointness.

Contribution

It establishes a proof of Emerton's conjecture, describes the right adjoint to parabolic induction explicitly, and applies these results to mod p Bernstein theory.

Findings

Proof of Emerton's conjecture on derived functors

Explicit description of Vignéras' right adjoint

Mod p Bernstein's Second Adjointness established

Abstract

We prove a variant of Emerton's conjecture concerning the right derived functors of the ordinary parts functor $\operatorname{Ord}_P^G$. This functor plays an important role in the theory of mod $p$ representations of $p$-adic reductive groups. A key ingredient for our proof is a comparison between certain small and parabolic inductions. Additionally, our method yields an explicit description of Vign\'eras' right adjoint to parabolic induction. In the appendix (joint with Heyer) we apply our results to obtain a mod $p$ variant of Bernstein's Second Adjointness, i.e. we show that the right and left adjoint of derived parabolic induction are isomorphic (on complexes with admissible cohomology) up to a cohomological shift and twist by a character.