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The connected components of affine Deligne--Lusztig varieties

Ian Gleason, Dong Gyu Lim, Yujie Xu

TL;DR

The paper resolves the general problem of determining the connected components of affine Deligne–Lusztig varieties at arbitrary parahoric levels by transferring the question to infinite-level moduli spaces of p-adic shtukas and exploiting v-sheaf techniques, Kimberlites, and specialization maps. Central to the approach is a p-adic Hodge–theoretic analysis of generic crystalline representations, which yields a Mumford–Tate criterion that precisely captures when the set of components is controlled by the Kottwitz–Kottwitz-phi invariants. The authors establish a robust circle of equivalences among bijectivity of component maps, HN-irreducibility, and the openness of derived images in crystalline representations, thereby proving the folklore conjecture in full generality (including non-quasisplit cases). Applications include new CM lifting results and p-adic uniformization statements for integral models of Shimura varieties at stabilizer-parahoric levels, linking local and global geometric structures via local Shimura varieties and shtuka moduli. The work unifies and extends prior cases, replacing ad hoc curve-connecting arguments with a cohesive framework grounded in Kimberlites and the Grothendieck–Messing period map, with broad implications for the geometry of Shimura varieties and their moduli spaces.

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.

The connected components of affine Deligne--Lusztig varieties

TL;DR

The paper resolves the general problem of determining the connected components of affine Deligne–Lusztig varieties at arbitrary parahoric levels by transferring the question to infinite-level moduli spaces of p-adic shtukas and exploiting v-sheaf techniques, Kimberlites, and specialization maps. Central to the approach is a p-adic Hodge–theoretic analysis of generic crystalline representations, which yields a Mumford–Tate criterion that precisely captures when the set of components is controlled by the Kottwitz–Kottwitz-phi invariants. The authors establish a robust circle of equivalences among bijectivity of component maps, HN-irreducibility, and the openness of derived images in crystalline representations, thereby proving the folklore conjecture in full generality (including non-quasisplit cases). Applications include new CM lifting results and p-adic uniformization statements for integral models of Shimura varieties at stabilizer-parahoric levels, linking local and global geometric structures via local Shimura varieties and shtuka moduli. The work unifies and extends prior cases, replacing ad hoc curve-connecting arguments with a cohesive framework grounded in Kimberlites and the Grothendieck–Messing period map, with broad implications for the geometry of Shimura varieties and their moduli spaces.

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.
Paper Structure (32 sections, 54 theorems, 132 equations)

This paper contains 32 sections, 54 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notations
  4. Main Results
  5. Rough Sketch of the argument
  6. More on the arguments
  7. Loop of the argument for \ref{['secondthm']}
  8. Proof for $(4)\implies(5)$ in \ref{['secondthm']}
  9. The Mumford-Tate group of "generic crystalline representations"
  10. Organization
  11. Group-theoretic setup
  12. Geometric background
  13. v-sheaf-theoretic setup
  14. $B^+_{dR}$-Grassmannians and local models
  15. Functoriality of affine Deligne--Lusztig varieties
  16. ...and 17 more sections

Key Result

Theorem 1.2

For all $p$-adic shtuka datum $(G,b,\mu)$ and all parahoric subgroups ${\mathcal{K}}_p\subseteq G({\mathbb Q}_p)$, mainconj holds.

Theorems & Definitions (113)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Remark 1.11
  • ...and 103 more