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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2022

Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$

Résumé

We consider the category of depth $0$ representations of a $p$-adic quasi-split reductive group with coefficients in $\overline{\mathbb{Z}}[\frac{1}{p}]$. We prove that the blocks of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters for $G$ over $\overline{\mathbb{Z}}[\frac{1}{p}]$. As a particular case, this depth $0$ category is thus indecomposable when the group is tamely ramified. Along the way we prove a similar result for finite reductive groups. We then outline a potential application to the Fargues-Scholze and Genestier-Lafforgue semisimple local Langlands correspondences. Namely, contingent on a certain "independence of $\ell$" property, our results imply that these correspondences take depth $0$ representations to tamely ramified parameters.

Dates et versions

hal-03903835 , version 1 (16-12-2022)

Identifiants

Citer

Jean-François Dat, Thomas Lanard. Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$. 2022. ⟨hal-03903835⟩
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