Modular cohomology of p-adic Deligne-Lusztig spaces

p-adic Deligne-Lusztig 空间的模上同调

• 批准号: 533474700
• 负责人: Professor Dr. Alexander Ivanov
• 金额: --
• 依托单位: Lehrstuhl Arithmetische Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/533474700?language=en
• 关键词: Modular; cohomology; adic; Deligne; Lusztig

The classical Deligne-Lusztig theory is a powerful geometric tool, which allows an essentially complete description of complex representations of finite groups of Lie type. It is also applicable to thestudy of modular representations of these groups (usually in non-defining characteristic). For example, recently it was used to prove Broué's abelian defect conjecture for such groups in many cases. Recently, the p-adic analogue of the classical Deligne-Lusztig theory became a focus of interest. It studies the so- called p-adic Deligne- Lusztig spaces. These are purely local and quite explicit objects, attached to p-adic reductive groups in a similar way as classical Deligne-Lusztig varieties are attached to finite groups of Lie type. The l-adic cohomology of p-adic Deligne-Lusztig spaces realizes many interesting representations of p-adic reductive groups to which they are attached, and allows to study these representations via methods from Deligne-Lusztig theory. However, their cohomology with mod l and l-adic integral coefficients was not studied yet. The aim of the present project is to fill in this gap in the theory, that is, to study the mod l and integral l-adic cohomology of p-adic Deligne-Lusztig spaces and, if possible, to apply it to the study of modular representations of p-adic reductive groups. Also, the relation of this cohomology with the local mod-l Langlands correspondences should be studied.

经典的Deligne-Lusztig理论是一个强大的几何工具，它允许对有限Lie型群的复表示进行基本完整的描述。它也适用于研究这些群的模表示（通常是非定义性的）。例如，最近它被用来证明Broué的交换亏损猜想，在许多情况下，这样的群体。最近，经典Deligne-Lusztig理论的p-adic模拟成为人们关注的焦点。它研究所谓的p-adic Deligne-Lusztig空间。这些都是纯粹的本地和相当明确的对象，附加到p-adic约化群以类似的方式作为经典Deligne-Lusztig品种附加到有限群的李型。p-adic Deligne-Lusztig空间的l-adic上同调实现了它们所连接的p-adic约化群的许多有趣的表示，并允许通过Deligne-Lusztig理论的方法来研究这些表示。然而，他们的上同调模l和l-adic整系数尚未研究。本项目的目的是填补这一空白的理论，即研究模l和积分l-adic上同调的p-adic Deligne-Lusztig空间，如果可能的话，将其应用到研究的模表示的p-adic约化群。此外，这种上同调与局部模-l Langlands对应的关系也需要研究。