#### Vol. 12, No. 4, 2018

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Parabolic induction and extensions

### Julien Hauseux

Vol. 12 (2018), No. 4, 779–831
##### Abstract

Let $G$ be a $p$-adic reductive group. We determine the extensions between admissible smooth mod $p$ representations of $G$ parabolically induced from supersingular representations of Levi subgroups of $G$, in terms of extensions between representations of Levi subgroups of $G$ and parabolic induction. This proves for the most part a conjecture formulated by the author in a previous article and gives some strong evidence for the remaining part. In order to do so, we use the derived functors of the left and right adjoints of the parabolic induction functor, both related to Emerton’s $\delta$-functor of derived ordinary parts. We compute the latter on parabolically induced representations of $G$ by pushing to their limits the methods initiated and expanded by the author in previous articles.

##### Keywords
$p$-adic reductive groups, mod p representations, parabolic induction, extensions, derived ordinary parts, Bruhat filtration
Primary: 22E50
##### Milestones
Revised: 23 April 2017
Accepted: 25 May 2017
Published: 11 July 2018
##### Authors
 Julien Hauseux Département de Mathématiques Université de Lille Villeneuve d’Ascq France