Parabolic induction and extensions

TL;DR

This paper characterizes the extension groups between certain smooth mod p representations of p-adic groups induced from supersingular representations, confirming a key conjecture and advancing the understanding of representation extensions.

Contribution

It determines extensions between parabolically induced representations from supersingular Levi subgroups, confirming a major conjecture and employing derived functors related to Emerton's delta-functor.

Findings

Extensions are explicitly computed in terms of Levi subgroup representations.

The work confirms most of a previously formulated conjecture.

Derived functors of parabolic induction are effectively used in the analysis.

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.