Complements sur les extensions entre series principales p-adiques et modulo p de G(F)

Abstract

We complete the results of a previous article. Let $G$ be a split connected reductive group over a finite extension $F$ of ${\mathbb{Q}}_p$. When $F={\mathbb{Q}}_p$, we determine the extensions between unitary continuous $p$-adic and smooth mod $p$ principal series of $G({\mathbb{Q}}_p)$ without assuming the centre of $G$ connected nor the derived group of $G$ simply connected. This shows a new phenomenon: there may exist several non-isomorphic non-split extensions between two distinct principal series. We also complete the computations of self-extensions of a principal series in the non-generic cases when the centre of $G$ is connected. Finally, we determine the extensions of a principal series of $G(F)$ by an "ordinary" representation of $G(F)$ (i.e. parabolically induced from a special representation twisted by a character). In order to do so, we compute Emerton's $\delta$-functor ${{\mathrm{H}}^{\bullet }\mathrm{Ord}}_{B(F)}$ of derived ordinary parts with respect to a Borel subgroup on an ordinary representation of $G(F)$.

33 pages, in French; minor correction in the proof of Lemma 2.2.3 in v2; added computations of self-extensions of principal series in non-generic cases in v3; added self-extensions of principal series for "irregular" characters and minor corrections in v4; typo corrected in Corollary 2.3.9 and updated references in v5 (final version)

References in corpus (1)

• On special representations of $p$-adic reductive groups

Cited by in corpus (3)

• Parabolic induction and extensions
• Multivariable $(\phi ,\Gamma )$-modules and smooth $o$-torsion representations
• p-adic etale cohomology of period domains