Affine Deligne-Lusztig Varieties of Positive Coxeter Type

We introduce a class of affine Deligne--Lusztig varieties that we call of positive Coxeter type. We show that the affine Deligne--Lusztig varieties of positive Coxeter type have a very simple and explicitly described geometric structure. Conversely, we explain how some of these geometric properties can be used to characterize this class. These results vastly generalize the work of He--Nie--Yu on affine Deligne-Lusztig varieties of finite Coxeter type, leading to applications to Shimura varieties that were not possible using the old notion.