Affine Deligne-Lusztig varieties are analogs (in the context of an affine root system) of Deligne-Lusztig varieties, which play an important role in the representation theory of finite groups of Lie type. They are subvarieties of affine Grassmannians or affine flag varieties, and studying them leads to combinatorial and geometric questions. Affine Deligne-Lusztig varieties serve to describe arithmetic properties of Shimura varieties, but they are also of purely group theoretic interest.
In this talk, I will explain the notion of affine Deligne-Lusztig variety, present it in a greater context, and report on new results (joint with X.
He) about their properties.