Lefschetz operator and local Langlands modulo $\ell$: the limit case

Abstract

Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with residue field ${\mathbb{F}}_q$, and let $\ell$ be a prime such that $q\equiv 1\left(mod\ell \right)$. We investigate the cohomology of the Lubin–Tate towers of $K$ with coefficients in ${\overline{\mathbb{F}}}_{\ell }$, and we show how it encodes Vignéras’ Langlands correspondence for unipotent ${\overline{\mathbb{F}}}_{\ell }$-representations.