Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917915395096576 |
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| author | Dat, Jean-François Lanard, Thomas |
| author_facet | Dat, Jean-François Lanard, Thomas |
| contents | 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. As an application, we deduce that the semi-simple local Langlands correspondence $π\mapsto φ_π$ constructed by Fargues and Scholze takes depth $0$ representations to tamely ramified parameters, using a motivic version of their construction recently announced by Scholze. We also bound the restriction of $φ_π$ to tame inertia in terms of the Deligne-Lusztig parameter of $π$ and show, in particular, that $φ_π$ is unramified if $π$ is unipotent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_03982 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$ Dat, Jean-François Lanard, Thomas Representation Theory 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. As an application, we deduce that the semi-simple local Langlands correspondence $π\mapsto φ_π$ constructed by Fargues and Scholze takes depth $0$ representations to tamely ramified parameters, using a motivic version of their construction recently announced by Scholze. We also bound the restriction of $φ_π$ to tame inertia in terms of the Deligne-Lusztig parameter of $π$ and show, in particular, that $φ_π$ is unramified if $π$ is unipotent. |
| title | Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2202.03982 |