论文标题
深度零表示超过$ \ overline {\ mathbb {z}}} [\ frac {1} {p} {p}] $
Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$
论文作者
论文摘要
我们考虑$ p $ -Adic准式分裂还原组的深度$ 0 $表示，其系数为$ \ overline {\ Mathbb {z}}} [\ frac {1} {p} {p}] $。我们证明，该类别的块是自然的，与$ g $ of $ \ overline {\ mathbb {z}} [\ frac {\ frac {1} {p} {p}] $ of $ g $ of $ g $ to $ g $ for $ g $ for $ g $的连接组件进行自然培养。因此，当小组被驯服时，此深度$ 0 $类别是不可塑性的。一路上，我们证明了有限还原组的类似结果。作为一个应用程序，我们推断出半简单的本地Langlands对应关系$π\ mapstoφ_π$由Fargues构建和Scholze构建，Scholze采用深度$ 0 $表示来驯服的参数，使用其最近由Scholze宣布的构造的动机版本。我们还将$φ_π$的限制限制在$π$的deligne-lusztig参数方面，尤其是表明，如果$φ_π$如果$π$是单一的。
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.
