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

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 $\mathrm{H^\bullet Ord}_{B(F)}$ of derived ordinary parts with respect to a Borel subgroup on an ordinary representation of $G(F)$.