Affine Deligne--Lusztig varieties are closely related to the mod-preduction of Shimura varieties and therefore play a central role in the Langlands program. In this talk, we will construct a tower of affine Deligne--Lusztig varieties for GLn and its inner forms. We prove that its limit at infinite level is isomorphic to the semi-infinite Deligne--Lusztig variety of Lusztig and that its cohomology realizes certain cases of automorphic induction and Jacquet--Langlands.
This is joint work with Alexander Ivanov.