仿射Deligne-Lusztig簇是 Rapoport-Zink 空间以群论的方式进行的推广，而 Rapoport-Zink空间作为 p-可除群的模空间是志村簇的局部类比，它的几何与志村簇特殊纤维的几何性质密切相关。本项目的主要目标是理解不分歧的仿射 Deligne-Lusztig 簇的几何性质，从而来理解志村簇特殊纤维上的牛顿分层。对于不分歧的仿射 Deligne-Lusztig 簇，申请人在与 E. Viehmann 教授的合作中在它上面定义了一个新的分层。在本项目中，我们希望了解这个分层的性质。同时对于几类特殊的仿射 Deligne-Lusztig 簇，我们希望理解它们的不可约分支的集合、每个不可约分支的刻画以及不同的不可约分支之间的相交情况。而且，我们还将探索仿射 Deligne-Lusztig 簇的几何与表示论之间的联系，以及几何 Satake 对应与弦拓扑之间的联系。

Affine Deligne-Lusztig varieties are the generalization of Rapoport-Zink spaces via group theoretic approach, while Rapoport-Zink spaces, as moduli spaces of p-divisible groups, are the local analogue of Shimura varieties. Its geometry is closely related to the special fiber of Shimura varieties. In this project, our aim is to study the geometry of unramified affine Deligne-Lusztig varieties and thus the Newton stratification of the special fiber of Shimura varieties. In a joint work with E. Viehmann(Munich Technical University), we have defined a new stratification on the affine Deligne-Lusztig varieties. In this project, we want to understand the properties of this stratification. And for some special affine Deligne-Lusztig varieties, we want to understand its set of irreducible components, the description of each irreducible component and the intersection behaviour of two irreducible components. Morevoer, we will investigate the relation between the geometry of affine Deligne-Lusztig varieties and representation theory, and also the relation between geometric Satake correspondence and string topology.