The connected components of affine Deligne--Lusztig varieties

Abstract

We compute the connected components of arbitrary parahoric level affine Deligne-Lusztig varieties for quasisplit reductive groups, 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.As an application, we deduce new CM lifting results on integral models of Shimura varieties at arbitrary parahoric levels. We also prove various results beyond the quasi-split case, by studying the cohomological dimensions of Newton strata inside the B + dR -affine Grassmannian. Contents 1. Introduction 1 Acknowledgements 10 2. Preliminaries and background 10 3. Generic Mumford-Tate groups 28 4. Proof of main theorems 31 5. Beyond the quasisplit case. 40 References 45

$2$-adic integral models of Shimura varieties with parahoric level structure

“…(1) Following [KZ21], we expect that the tame condition on G can be relaxed. (2) We expect that the results in [Hof20] and [GLX22] can be extended to the 2-adic models constructed in this paper.…”

On the connectedness of $p$-adic period domains

“…Our work together with [GLX22] finishes the problem of computing connected components of ADLV in mixed characteristic.…”

“…This is where the connected components of ADLV enter in her argument. In [GLX22], the first author together with Lim and Xu show that Chen's reasoning can be reversed, and use Theorem 1.1 to compute the connected components of ADLV and the connected components of LSV ([GLX22, Theorem 1.14, Theorem 1.2. (3)]).…”