The connected components of affine Deligne--Lusztig varieties

TL;DR

This paper determines the connected components of affine Deligne-Lusztig varieties and local Shimura varieties, resolving a long-standing conjecture using advanced p-adic and v-sheaf techniques, with applications to Shimura varieties and CM lifting.

Contribution

It provides a complete computation of connected components for affine Deligne-Lusztig varieties and local Shimura varieties, including non-quasisplit groups, using v-sheaf methods.

Findings

Resolved a folklore conjecture on connected components.

Connected components characterized via p-adic Hodge theory.

New results on integral models of Shimura varieties and CM lifting.

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.