本课题研究仿射Deligne-Lusztig簇基本的几何拓扑性质.这些性质在志村簇模p约化的研究中扮演着重要的角色.研究内容主要分以下两个方面:(1)给出仿射旗簇中的仿射Deligne-Lusztig簇的非空判定法则并确定它们的维数;(2)对仿射Grassmann簇中的仿射Deligne-Lusztig簇的不可约分支进行分类,并研究它们与几何表示论中的联系.在此基础上,对仿射Deligne-Lusztig簇(及其闭包)的连通分支进行分类.

This is a proposal on basic geometric and topological properties of affine Deligne-Lusztig varieties. These properties play an important role in the study of the reduction modulo p of Shimura varieties. The proposal consists of two major parts as follows: (1) Provide a criterion for the non-emptiness of affine Deligne-Lusztig varieties in an affine flag varieties, and determine their dimensions; (2) Classify the irreducible components of affine Deligne-Lusztig varieties in an affine Grassmannian, and study their connections with geometric representation theory. Based on this classification, classify the connected components of affine Deligne-Lusztig varieties (and their closures).