Rev\^etements du demi-plan de Drinfeld et Langlands p-adique cat\'egorique

TL;DR

This paper extends the categorification of the p-adic local Langlands correspondence to all levels of Drinfeld's tower, introduces new functors, and analyzes associated sheaves, revealing finite length properties of certain quotients.

Contribution

It generalizes an exact sequence to all levels of Drinfeld's tower and introduces two categorified functors in Banach and locally analytic settings.

Findings

Extended exact sequence to all levels of Drinfeld tower

Introduced two categorification-inspired functors

Proved finite length property for quotients of supercuspidal representations

Abstract

We generalize to all levels of the tower of coverings of the Drinfeld upper plane an exact sequence established by Lue Pan for the first covering. Furthermore, we introduce two functors, inspired by the categorification of the $p$-adic local Langlands correspondence, in Banach and locally analytic versions respectively. We then compute the sheaves associated with the representations appearing in our sequence. As an application, we show that all proper quotients of the universal unitary completion of a supercuspidal representation have finite length.