publications

2022

1. The connected components of affine Deligne-Lusztig varieties

   Ian Gleason, Dong Gyu Lim, and Yujie Xu

   2022

   Abs arXiv

   We compute the connected components of arbitrary parahoric level affine Deligne–Lusztig varieties and local Shimura varieties, thus resolving the folklore conjecture in full generality (including non-quasisplit groups). We achieve this by relating them to the connected components of infinite level moduli spaces of p-adic shtukas, where we use v-sheaf-theoretic techniques such as the specialization map of kimberlites. Along the way, we give a p-adic Hodge-theoretic characterization of HN-irreducibility.
   As applications, we obtain many results on the geometry of integral models of Shimura varieties at arbitrary parahoric levels. In particular, we deduce new CM lifting results on integral models of Shimura varieties for quasi-split groups at arbitrary connected parahoric levels.

2023

1. Nonemptiness criterion of affine Deligne-Lusztig varieties

   Dong Gyu Lim

   2023

   Abs arXiv

   Affine Deligne-Lusztig varieties with various level structures show up in the study of Shimura varieties and moduli spaces of shtukas. Among is the Iwahori level structure which is the most refined one. We study the nonemptiness problem of single affine Deligne-Lusztig varieties at Iwahori level in the basic case. Under a genericity condition (the “shrunken Weyl chambers” condition), an explicit criterion is known. However, no explicit criterion has been available without the condition even conjecturally. We conjecture a new criterion in full generality, and prove it except for finitely many cases. As an application, the nonemptiness problem in some non-basic cases and a new conjectural dimension formula are discussed.

2. A combinatorial proof of the general identity of He-Nie-Yu

   Dong Gyu Lim

   2023

   Abs arXiv

   We give a uniform and combinatorial proof of the general identity appearing in the work of He-Nie-Yu on the affine Deligne-Lusztig varieties with finite Coxeter part.