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

TL;DR

This paper completes the classification of extensions between principal series representations of split reductive groups over local fields, revealing new phenomena and computing derived functors related to ordinary parts.

Contribution

It extends previous work by determining all extensions between principal series over F, including non-split and non-generic cases, and computes derived functors of ordinary parts.

Findings

Existence of multiple non-isomorphic non-split extensions between principal series.

Complete description of self-extensions in non-generic cases.

Extension computations between principal series and ordinary representations.

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