[Paper Review] Affine Deligne-Lusztig varieties and the action of J

Miaofen Chen, Eva Viehmann|arXiv (Cornell University)|Jul 10, 2015

Advanced Algebra and Geometry15 references4 citations

TL;DR

This paper introduces a new group-theoretic stratification of affine Deligne-Lusztig varieties using the action of the group $J_b(F)$, defined via the relative position function $f_g: J_b(F) \to X_*(T)_{\text{dom}}$. The key contribution is a unified interpretation of known stratifications—such as Bruhat-Tits, semi-module, and $a$-invariant stratifications—as special cases of this general framework, with the $a$-invariant equal to 1 if and only if $f_g$ lies in a specific $J_b(F)$-orbit.

ABSTRACT

We propose a new stratification of the reduced subschemes of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties that highlights the relation between the geometry of these spaces and the action of the associated automorphism group. We show that this provides a joint group-theoretic interpretation of well-known stratifications which only exist for special cases such as the Bruhat-Tits stratification of Vollaard and Wedhorn, the semi-module stratification of de Jong and Oort, and the locus where the a-invariant is equal to 1.

Motivation & Objective

• To unify disparate stratifications of affine Deligne-Lusztig varieties—such as Bruhat-Tits, semi-module, and $a$-invariant stratifications—into a single group-theoretic framework.
• To clarify the geometric role of the group $J_b(F)$ in the structure of these varieties by defining a new invariant based on relative positions.
• To provide a conceptual explanation for the observed link between the geometry of affine Deligne-Lusztig varieties and the Bruhat-Tits building of $J_b(F)$, which had previously lacked a general formulation.
• To show that the $a$-invariant being 1 corresponds precisely to the $J_b(F)$-orbit of a canonical function $f_g$, thus giving a group-theoretic characterization of this invariant.

Proposed method

• Define a function $f_g: J_b(F) \to X_*(T)_{\text{dom}}$ for each $g \in G(L)/K$ via the Cartan decomposition, assigning to each $j \in J_b(F)$ the relative position $\text{inv}(j,g)$.
• Use the Cartan decomposition $G(L) = \coprod_{\xi \in X_*(T)_{\text{dom}}} K\xi(\epsilon)K$ to compute the value of $f_g(j)$ as the cocharacter corresponding to $j^{-1}g \in K\epsilon^{f_g(j)}K$.
• Construct a stratification of $X_\mu(b)$ into locally closed subschemes indexed by the image of $f_g$, showing that this stratification generalizes known cases.
• Prove that the $a$-invariant of a lattice $M = gM_0$ equals 1 if and only if $f_g$ lies in the $J_b(F)$-orbit of a fixed function $F$, using the multi-semimodule structure of Dieudonné modules.
• Use the action of $J_b(F)$ to normalize lattices $M \subset M_0$ and reduce to the case where $M$ is not contained in any $jM_0$ with $v_\epsilon(\det j) > 0$, simplifying the analysis of the $a$-invariant.
• Leverage the structure of isocrystals and the action of $\tau_0 = \sigma$ and $\tau_1 = (b\sigma)^a \epsilon^{a'}$ to characterize $J_b(F)$-stable lattices via invariance under these operators.

Experimental results

Research questions

• RQ1 How can the geometry of affine Deligne-Lusztig varieties be systematically related to the action of the group $J_b(F)$?
• RQ2 Can the Bruhat-Tits stratification, semi-module stratification, and $a$-invariant stratification be understood as special cases of a single, unified group-theoretic stratification?
• RQ3 What is the precise group-theoretic condition on $f_g$ that characterizes when the $a$-invariant of a lattice $M = gM_0$ is equal to 1?
• RQ4 Is the closure of a stratum in the new stratification always a union of strata, and if not, what is the nature of the closure relations?

Key findings

• The new stratification of $X_\mu(b)$ via the function $f_g: J_b(F) \to X_*(T)_{\text{dom}}$ is locally closed and generalizes known stratifications such as the Bruhat-Tits stratification and the semi-module stratification.
• The $a$-invariant of a lattice $M = gM_0$ is equal to 1 if and only if $f_g$ lies in the $J_b(F)$-orbit of a canonical function $F$, providing a group-theoretic characterization of this invariant.
• For lattices $M \subset M_0$ not contained in any $jM_0$ with $v_\epsilon(\det j) > 0$, the multi-semimodule of $M$ equals $A_{\text{gen}}$ if and only if $a(M) = 1$, and this condition is equivalent to $f_g$ being in the specified $J_b(F)$-orbit.
• The stratification is not necessarily closed under taking closures, as shown by a counterexample in Section 2.1, indicating that the closure relations are non-trivial and not simply unions of strata.
• The action of $J_b(F)$ on $X_\mu(b)$ allows for a normalization of $g$ such that $M = gM_0 \subset M_0$ and $M$ is not contained in any $jM_0$ with $v_\epsilon(\det j) > 0$, simplifying the analysis of the $a$-invariant.
• A lattice $M$ in the isocrystal $(L^h, b\sigma)$ is $J_b(F)$-stable (i.e., $M = jM_0$ for some $j \in J_b(F)$) if and only if it is stable under the operators $\tau_0$ and \tau_1$, which generate the action of $b\sigma$ and $\epsilon$.