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Nonemptiness of single affine Deligne-Lusztig varieties

This paper proposes and largely proves a new explicit criterion for the nonemptiness of single affine Deligne-Lusztig varieties at Iwahori level in the basic case, removing the previously required "shrunken Weyl chambers" genericity condition.

Original authors: Dong Gyu Lim

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Dong Gyu Lim

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to build a massive, intricate city called Shimura City. This city is built on a very strange, shifting foundation made of numbers and symmetries. To understand how the city is laid out, you need to know which specific plots of land (mathematical spaces) are actually buildable (non-empty) and which are just empty holes in the ground.

These specific plots are called Affine Deligne-Lusztig Varieties.

For a long time, mathematicians had a rulebook for deciding if a plot was buildable, but it only worked if the plot was in a very specific, "safe" neighborhood of the city. If the plot was in a tricky, borderline area, the rulebook was silent. This paper, by Dong Gyu Lim, is like a new, universal map that finally tells us exactly which plots are buildable, even in those tricky areas.

Here is the story of the paper, broken down into simple concepts:

1. The City and the Plots

Think of the city as a giant, multi-dimensional grid.

The Grid: This is the "Iwahori-Weyl group." It's like a massive chessboard where every square represents a possible configuration of the city.
The Plots: Each square on the board is a "Single Affine Deligne-Lusztig Variety."
The Goal: We want to know: "Is this specific square occupied by a building, or is it empty?"

2. The Old Rulebook (The "Shrunken" Neighborhood)

Previously, mathematicians (specifically He, Görtz, and Rapoport) figured out a perfect rule for a large, safe zone of the city called the "Shrunken Weyl Chambers."

The Rule: If your plot is in this safe zone, it's buildable if and only if it passes a specific "symmetry test" (checking if the plot's shape matches the city's overall symmetry).
The Problem: The city is huge. There are "Critical Strips"—narrow, dangerous corridors running through the city where the old rulebook didn't work. These strips are like the foggy edges of a map where the rules get blurry. No one knew for sure if plots in these strips were buildable or not.

3. The New Discovery: The "Embedding" Trick

Lim's big idea is to stop looking at the dangerous strips as isolated, confusing places. Instead, he treats them as overlapping neighborhoods.

Imagine you are standing in a foggy corridor (a Critical Strip). You can't see the whole city, but you know that this corridor touches two different "Safe Zones" (Shrunken Weyl Chambers) on either side.

The Analogy: Lim says, "Let's pretend we are standing in the Safe Zone on the left, check the rule. Then, let's pretend we are standing in the Safe Zone on the right, and check the rule again."
The Set $W_x$ : He creates a special list of "perspectives" (mathematically called the set $W_x$ ). For a plot in a safe zone, you only need one perspective. But for a plot in a critical strip, you need to check multiple perspectives to see if the plot holds up.

4. The New Universal Rule

Lim proposes a new, all-encompassing rule:

"A plot is buildable if, and only if, it passes the symmetry test from every single perspective in your special list ( $W_x$ )."

The "If" part: He proves that if a plot passes the test from all perspectives, it is definitely buildable.
The "Only If" part: He proves that if it fails even one perspective, it is empty.
The Catch: He proves this works for almost every plot in the city. There are a tiny, finite number of weird, edge-case plots where the proof is still being finalized, but for 99.9% of the city, the map is complete.

5. Why This Matters

Why do we care if a mathematical plot is empty or not?

The Blueprint: These plots are the blueprints for understanding Shimura Varieties, which are crucial for solving deep problems in number theory (like the famous Langlands Program).
The Dimensions: Knowing a plot exists allows mathematicians to calculate its "size" (dimension). Lim's work helps create new formulas to calculate the size of these spaces, even in the foggy critical strips where it was previously impossible.
The "Cordial" Elements: He also identifies a special class of "friendly" plots (called cordial elements) where the rules are extra simple, allowing for a complete catalog of all possible buildings in those areas.

Summary

Think of this paper as the final expansion pack for a video game map.

Before: You could explore the safe, sunny plains, but the foggy borderlands were a mystery. You didn't know if there were treasures (buildings) there or just void.
Now: Lim has handed you a compass and a new set of rules. He says, "If you look at the borderland from these specific angles, you can see exactly where the treasures are."

This work removes the guesswork, providing a clear, explicit criterion for mathematicians to navigate the complex, shifting landscape of modern number theory.

1. Problem Statement

The paper addresses the nonemptiness problem for single affine Deligne-Lusztig varieties (ADLVs) at the Iwahori level.

Context: ADLVs, denoted $X_x(b)$ , arise naturally in the study of the special fibers of Shimura varieties and moduli spaces of shtukas. They are defined as subsets of the affine flag variety $G(\breve{Q}_p)/I$ (where $I$ is an Iwahori subgroup) given by:
$X_x(b) := \{gI \in G(\breve{Q}_p)/I : g^{-1}b\sigma(g) \in I\dot{x}I\}$
where $b \in G(\breve{Q}_p)$ , $x$ is an element of the Iwahori-Weyl group $\tilde{W}$ , and $\sigma$ is the Frobenius automorphism.
The Gap: While the nonemptiness of general ADLVs (with parahoric level structures) is well-understood via Mazur's inequality (i.e., $X(\mu, b)_K \neq \emptyset \iff [b] \in B(G, \mu)$ ), the nonemptiness of single ADLVs (where the level is the finest Iwahori structure) is more subtle.
Previous State of Knowledge:
- A criterion exists for $x$ lying in "shrunken Weyl chambers" (regions of the building far from the base alcove), established by Görtz, He, and Nie (GHN15). This criterion relies on the $\sigma$ -support of an element $\eta_\sigma(x)$ .
- However, no explicit criterion existed for $x$ lying in "critical strips" (regions adjacent to the base alcove), even conjecturally. The existing general criterion (GHN A) is abstract and difficult to compute in practice.

2. Methodology and New Ideas

Lim introduces a new geometric and combinatorial framework to extend the nonemptiness criterion beyond the shrunken Weyl chambers.

The Set $W_x$ : The core innovation is the definition of a subset $W_x \subset W_0$ $W_{x} \subset W_{0}$ (the finite Weyl group). Intuitively, if an element $x$ $x$ lies in a critical strip, it "belongs" to the adjacent shrunken Weyl chambers. $W_x$ $W_{x}$ consists of elements $r \in W_0$ $r \in W_{0}$ that effectively "embed" $x$ $x$ into these adjacent chambers.
- If $x$ is in a shrunken chamber, $W_x = \{id\}$ .
- If $x$ is in exactly one critical strip, $W_x = \{id, s\}$ where $s$ is the simple reflection associated with that strip.
Anti-Closed Root Sets: The author analyzes the set of positive roots $\Phi_x$ such that $x$ lies in the critical strip associated with $\alpha$ . It is shown that the complement $\Phi^+ \setminus \Phi_x$ is a radical closed subset of the root system. This structural property allows for the rigorous definition of $W_x$ and the proof that it captures all necessary obstructions.
Connection to Length-Positive Sets: The paper establishes a relationship between $W_x$ and the "length-positive" set $LP(x)$ defined by Felix Schremmer, showing they are closely related via the element $v_x$ (the finite part of $x$ ).
Strategy: The proof strategy generalizes the "No Levi Obstruction" (NLO) principle from GHN15. Instead of checking a single condition, the author proves that $X_x(b) \neq \emptyset$ if and only if the "No Levi Obstruction" condition holds for all embeddings of $x$ into adjacent chambers, indexed by $r \in W_x$ .

3. Key Contributions and Main Results

A. The Main Conjecture (Conjecture 1.2)

The paper proposes a unified, explicit criterion for the nonemptiness of $X_x(b)$ in the basic case (where $b$ is basic).
Conjecture: Let $G$ be simple, quasi-split, and adjoint. Let $b$ be basic and $\kappa_G(x) = \kappa_G(b)$ . Assume $\widehat{\text{supp}}_\sigma(x) = \tilde{S}$ (a technical condition ensuring $x$ is not "too central").
Then, $X_x(b) \neq \emptyset$ if and only if:
$\text{supp}_\sigma(\sigma^{-1}(r)\eta_\sigma(x)r^{-1}) = S \quad \text{for all } r \in W_x$
where $S$ is the set of simple reflections.

B. Main Theorem (Theorem 1.3)

The author proves the conjecture in full generality for the "if" direction and for "all but finitely many" cases for the "only if" direction.

General "If" Direction: The condition is sufficient for all $x$ .
General "Only If" Direction: The condition is necessary for all $x$ with sufficiently large length ( $\ell(x) \gg 0$ ).
Specific Proven Cases: The "only if" direction is proven for:
- $x$ lying in exactly one critical strip.
- Translation elements ( $x = t_\mu$ ) and $vt_\mu$ -form elements.
- Type $A_n$ groups under specific coefficient conditions on $\mu$ .

C. Stronger Version of GHN A (Theorem 1.6)

For elements with large length ( $\ell(x) \gg 0$ ) and $\widehat{\text{supp}}_\sigma(x) = \tilde{S}$ , the paper proves a stronger, more geometric criterion:
$X_x(b) \neq \emptyset$ if and only if $x$ is not a $(J, w)^\sigma$ -alcove for any proper $\sigma$ -stable subset $J \subsetneq S$ .
This implies that for large $x$ , nonemptiness fails if and only if $x$ "comes from" a proper Levi subgroup.

D. Applications

Cordial Elements: The paper applies the new criterion to describe the set $B(G)_x$ (the set of $\sigma$ -conjugacy classes intersecting $I\dot{x}I$ ) for cordial elements. It shows that for certain translation elements, $B(G)_x$ is saturated and equals the set of classes satisfying Mazur's inequality.
Dimension Formulas: The author discusses a new conjectural dimension formula for $x$ in a single critical strip (Theorem 1.10), which extends the known formula for shrunken chambers.

4. Significance

Resolution of a Longstanding Gap: This work removes the restrictive "shrunken Weyl chambers" condition that has limited the applicability of explicit nonemptiness criteria for ADLVs for over a decade.
Explicit Computability: The new criterion replaces the abstract "Levi obstruction" check (which requires iterating over all possible Levi subgroups) with a finite check over the set $W_x$ , which is small and explicitly computable based on the position of $x$ .
Foundational for Future Work: By establishing the nonemptiness pattern for critical strips, the paper provides the necessary foundation for deriving dimension formulas and understanding the geometry of Rapoport-Zink spaces and local Shimura varieties in previously inaccessible regions.
Complementary Results: The work complements recent developments by Felix Schremmer on generic Newton points, linking the combinatorial structure of the Iwahori-Weyl group to the geometry of the building.

In summary, Dong Gyu Lim provides a comprehensive and largely proven conjecture that fully characterizes the nonemptiness of single affine Deligne-Lusztig varieties at the Iwahori level, bridging the gap between the well-understood "shrunken" regions and the critical "strip" regions of the Bruhat-Tits building.