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arXiv · 2208.07195

The connected components of affine Deligne--Lusztig varieties

Ian Gleason·Dong Gyu Lim·Yujie Xu
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Abstract

We compute the connected components of arbitrary parahoric level affine Deligne-Lusztig varieties and local Shimura varieties, thus resolving a folklore conjecture in full generality (even for 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 of Hodge type at arbitrary stabilizer-parahoric levels. In particular, we deduce new CM lifting results on integral models of Shimura varieties for quasisplit groups at parahoric levels that arise as stabilizer Bruhat-Tits group schemes.

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Ian Gleason, Dong Gyu Lim, Yujie Xu. 2022-08-15. The connected components of affine Deligne--Lusztig varieties. https://doi.org/10.1007/s00222-025-01386-1

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