Sur une conjecture de Breuil-Herzig

Abstract

Let $G$ be a split $p$-adic reductive group with connected centre and simply connected derived subgroup. We show that certain "chains" of principal series of $G$ do not exist and we establish several properties of the Breuil-Herzig construction $\Pi (\rho {)}^{\mathrm{ord}}$. In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil-Herzig. In order to do so, we partially compute Emerton's $\delta$-functor ${{\mathrm{H}}^{\bullet }\mathrm{Ord}}_P$ of derived ordinary parts with respect to a parabolic subgroup on a principal series. We formulate a new conjecture on the extensions between smooth mod $p$ representations of $G$ parabolically induced from supersingular representations of Levi subgroups of $G$ and we prove it in the case of extensions by a principal series.

35 pages, in French; minor changes in v2; major improvement of the main results in v3; article shortened (sections 2 and 3 merged), minor changes in section 4, conjecture 4.4.3 corrected in v4

References in corpus (1)

• Extensions entre series principales p-adiques et modulo p de G(F)

Cited by in corpus (5)

• Parabolic induction and extensions
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• On the exactness of ordinary parts over a local field of characteristic $p$
• Towards the finite slope part for ${\mathrm{GL}}_n$
• p-adic etale cohomology of period domains