Representations of Affine Hecke Algebras and Geometry of Shimura Varieties

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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.