Modular cohomology of p-adic Deligne-Lusztig spaces

点击翻译按钮获取中文摘要

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.