Parabolic induction and extensions

Julien Hauseux

Published 2016-07-07Version 1

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 parabolic induction, 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. These computations will also have applications to the deformation theory of parabolically induced smooth mod $p$ representations of $G$ in a forthcoming joint article with T. Schmidt and C. Sorensen.