Sur une conjecture de Breuil-Herzig

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 $\mathrm{H^\bullet Ord}_P$ 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.