Parabolic induction and extensions

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.

arXiv comment55 pages, final version

References in corpus (8)

• The classification of irreducible admissible mod p representations of a p-adic GL_n
• A classification of irreducible admissible mod p representations of p-adic reductive groups
• Ordinary representations of G(Q_p) and fundamental algebraic representations
• Une étude des représentations modulo $p$ de SL(2,F)
• On a classification of irreducible admissible modulo $p$ representations of a $p$-adic split reductive group
• Extensions entre series principales p-adiques et modulo p de G(F)
• Sur une conjecture de Breuil-Herzig
• Complements sur les extensions entre series principales p-adiques et modulo p de G(F)

Cited by in corpus (9)

• Functorial properties of generalised Steinberg representations
• On the exactness of ordinary parts over a local field of characteristic $p$
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• Functorial properties of pro-$p$-Iwahori cohomology
• Deformation rings and parabolic induction
• p-adic etale cohomology of period domains
• The Left Adjoint of Derived Parabolic Induction
• On Extensions of supersingular representations of ${\mathrm{SL}}_2({\mathbb{Q}}_p)$
• Derived right adjoints of parabolic induction: an example