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

TL;DR

This paper computes extensions between certain p-adic and mod p principal series representations of split reductive groups over p-adic fields, advancing understanding in the p-adic and mod p Langlands programs.

Contribution

It determines extensions between unitary continuous p-adic and smooth mod p principal series representations in the generic case, using derived functors and Bruhat filtrations.

Findings

Computed Emerton's delta-functor for specific induced representations.

Determined extension groups relevant to p-adic and mod p Langlands correspondences.

Provided explicit descriptions of extensions in the generic case.

Abstract

Let $G$ be a split connected reductive group over a finite extension $F$ of $Q_p$. We determine the extensions between unitary continuous $p$-adic and smooth mod $p$ principal series of $G(F)$ in the generic case. 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 certain induced representations of $G(F)$ using a Bruhat filtration. These extensions come into play in the $p$-adic and mod $p$ Langlands programs.