A combinatorial study of affine Deligne-Lusztig varieties

Abstract

We consider affine Deligne-Lusztig varieties $X_w(b)$ and certain unions $X(\mu,b)$ in the affine flag variety of a connected reductive group. They were first introduced by Rapoport to facilitate the study of mod-$p$ reduction of Shimura varieties and moduli spaces of shtukas. We improve upon certain existing results in the study of affine Deligne-Lusztig varieties by weakening the hypothesis to prove them. Such results include a description of generic Newton points in Iwahori double cosets in the loop group of a split reductive group, covering relations in the associated Iwahori-Weyl group, and a dimension formula for $X(\mu,b)$ in the case of a quasi-split group. As an application of the work on generic Newton point formula, we obtain a description of the dimension for $X(\mu,b)$ associated with the maximal element $b$ in its natural range, under a mild hypothesis on $\mu$ but no further restrictions on the group.