Representations of Affine Hecke Algebras and Geometry of Shimura Varieties

人们在日常生活中会遇到对称：它出现在人脸、蝴蝶、雪花等。在数学和自然科学中，对称是丰富的，也有多种形式。Weyl群是当前数学研究的一个引人入胜的领域，它是由满足辫状关系的自反对称组成的某些对称的集合。Weyl群结构的研究在理解描述物理系统对称性的更大群的表示和志村变的几何中起着基础作用，这是现代数学的两个中心主题。在这项研究中，PI将研究Weyl群的对称性及其在表示理论和算术几何中的应用。首席研究员将利用最近关于Weyl群的发现来研究仿射Hecke代数的表示以及仿射delign - lusztig变体和Shimura变体的几何。最近，PI在仿射Hecke代数和仿射delign - lusztig变体之间建立了联系，为表示理论和算术几何之间提供了新的桥梁。在这个项目中，PI将系统地发展这一想法，并将其应用于仿射Hecke代数的模表示的研究和志村品种的特殊纤维的研究。

People encounter symmetry in daily life: it appears in human faces, butterflies, snowflakes, etc. In mathematics and natural science, symmetry is abundant and takes many forms as well. A fascinating area of current mathematical study concerns Weyl groups, which are certain collections of symmetries obtained by compositions of reflexive symmetries that satisfy braid relations. The study of the structure of the Weyl groups plays a fundamental role in understanding representations of the much larger groups that describe the symmetries of physical systems and in the geometry of Shimura varieties, two central themes in modern mathematics. In this research, PI will study the symmetries of Weyl groups and their applications in representation theory and arithmetic geometry.The principal investigator will use recent discoveries about Weyl groups to study the representations of affine Hecke algebras and the geometry of affine Deligne-Lusztig varieties and Shimura varieties. Recently, the PI established a connection between the affine Hecke algebras and the affine Deligne-Lusztig varieties that provides a new bridge between representation theory and arithmetic geometry. In this project, the PI will systematically develop this idea and will apply it to the study of modular representations of affine Hecke algebras and the study of special fibers of Shimura varieties.