Arithmetic Kashiwara Regularity and Orbit Classification for Filtered Strongly Equivariant $\mathcal{D}^{\dagger}$-Modules

Technical Summary: Arithmetic Kashiwara Regularity and Orbit Classification for Filtered Strongly Equivariant $D^\dagger$-Modules

Problem Statement
The paper addresses the arithmetic analogue of Kashiwara's regularity and orbit classification theorem for equivariant $D$-modules on flag varieties. In the classical complex setting, coherent equivariant $D$-modules on flag varieties are regular holonomic, and simple objects are classified by pairs consisting of a group orbit and an irreducible equivariant local system. The goal is to establish a corresponding theory for Berthelot's arithmetic differential operators ($D^\dagger$-modules) on formal flag varieties over a complete discrete valuation ring of mixed characteristic $(0, p)$.

A central difficulty in the arithmetic setting is that "holonomicity" alone is insufficient to replace "regular holonomicity." The stable category for the six operations is the category of overholonomic modules. Furthermore, the classical proof relies on the compatibility of the group action with infinitesimal symmetries (strong equivariance) to force the characteristic variety into the conormal bundle of the orbits. In the arithmetic context, the characteristic variety is defined via finite-level models, requiring a precise notion of equivariance that is visible at these finite levels.

Methodology
The author develops a framework based on filtered strong equivariance for coherent $D^\dagger$-modules. The methodology proceeds through the following logical steps:

1. Filtered Strong Equivariance: The paper introduces a definition of $K$-equivariance that combines a weak linearization (global symmetry) with two stronger conditions:

   • Infinitesimal Compatibility (SE1): The infinitesimal action induced by the linearization must coincide with the action of fundamental vector fields inside $D^\dagger$.
   • Filtered Finite-Level Realizability (SE2): This coincidence must be realized on sufficiently large good finite-level models ($\widehat{\mathcal{D}}^{(m)}$) with respect to good filtrations. This ensures that the principal symbols of the fundamental vector fields are well-defined and vanish on the associated graded module.

2. Microlocal Analysis (Moment Map Containment): Using the finite-level models, the author constructs an arithmetic moment map $\mu_K: T^*X_s \to \mathfrak{k}_s^\vee$. The core microlocal argument demonstrates that for any filtered strongly equivariant module $M$, the principal symbols of all fundamental vector fields vanish on the characteristic variety $\text{Car}(M)$. Consequently, $\text{Car}(M)$ is contained in the zero fiber of the moment map, which, under the hypothesis of orbit separability, equals the union of the conormal bundles to the $K_s$-orbits.

3. Holonomicity and Overholonomicity:

   • Holonomicity: The conormal containment, combined with the assumption of finitely many orbits and the arithmetic Bernstein inequality, forces $\dim \text{Car}(M) = \dim X_s$, proving that $M$ is holonomic. This result holds without assuming a Frobenius structure.
   • Geometric Overholonomicity: In the presence of a Frobenius structure, the author invokes Caro's resolution of Berthelot's holonomicity-stability conjectures for smooth projective formal schemes. This upgrades Frobenius holonomicity to Frobenius overholonomicity. Combined with Huyghe–Schmidt's base-change conventions, this yields geometric overholonomicity.

4. Orbital Devissage: Once objects are placed in the overholonomic heart, the paper performs an equivariant orbital devissage. By restricting to open orbits and using arithmetic intermediate extensions ($j_{!+}$), simple objects are decomposed into pieces supported on single orbits.

Key Contributions and Results

• Theorem 1.1 (Arithmetic Kashiwara Regularity):

  • Every filtered strongly $K$-equivariant coherent $D^\dagger_{X, \mathbb{Q}}$-module is holonomic.
  • Every Frobenius filtered strongly $K$-equivariant coherent $D^\dagger_{X, \mathbb{Q}}$-module is geometrically overholonomic.
  • Significance: This establishes that the "filtered strong" condition is a regularity mechanism. It forces objects into the holonomic and overholonomic ranges based on orbit geometry, rather than assuming overholonomicity a priori.

• Theorem 1.2 (Filtered Arithmetic Orbit Classification):

  • Simple objects in the category of Frobenius filtered strongly equivariant coherent $D^\dagger$-modules are classified by pairs $(O, E)$, where:
    • $O$ is a $K_s$-orbit in the special fiber $X_s$.
    • $E$ is an irreducible $K$-equivariant overconvergent $F$-isocrystal on $(O, O)$.
  • The classification map is given by restriction to the open orbit, and the inverse is the arithmetic intermediate extension $j_{O, !+}(E)$.
  • Crucially, the image of this classification is restricted to those isocrystals whose intermediate extensions satisfy the filtered strong equivariance condition (denoted as the admissible locus $\text{Adm}^{\Phi, \text{fstr}}_K$).

• Corollary 1.1 (Crystalline Distribution Algebra):

  • Via the arithmetic Beilinson–Bernstein localization at the trivial character, the geometric classification is transported to an algebraic classification. Simple Frobenius filtered strongly equivariant finitely presented modules over the crystalline distribution algebra $A_0$ are classified by the same orbit-isocrystal pairs.

Significance and Claims
The paper claims to provide the "arithmetic analogue of the regularity part of Kashiwara's theorem." Its primary contribution is the equivariant finite-level regularity mechanism.

• Regularity vs. Existence: The author explicitly states that this is a regularity theorem, not an existence theorem. It proves that if an object satisfies the filtered strong condition, it is forced into the holonomic/overholonomic range. It does not claim that the filtered strong condition is automatic for all weakly equivariant objects, nor does it claim that the resulting class of modules is larger than those obtained by restricting non-equivariant overholonomic classifications to orbit supports.
• Microlocal Necessity: The "filtered" clause is not a cosmetic addition but a technical necessity. In Berthelot's theory, characteristic varieties are defined via finite-level models. The strong condition ensures that the infinitesimal symmetry is visible at this level, allowing the principal symbol argument to function. Without this, large coherent modules (like $D^\dagger$ itself) could carry weak equivariance but fail to be holonomic.
• Distinction from Previous Work: While Huyghe and Schmidt previously classified irreducible modules using overconvergent isocrystals on locally closed subvarieties, this paper refines that picture to the $K_s$-orbit stratification. It proves that under the finite-orbit and orbit-separability hypotheses, the support of a simple equivariant object is exactly the closure of a single orbit, and the coefficient is an equivariant isocrystal on that orbit.

The paper concludes by noting that the classification relies on specific hypotheses: the subgroup $K$ must be smooth, the action on the special fiber must have finitely many orbits, and the orbit-separability condition must hold (ensuring the infinitesimal action detects tangent directions). The results are presented as a model for a broader theory of filtered strongly equivariant arithmetic $D^\dagger$-modules on stratified formal schemes, though the current scope is limited to formal flag varieties at the trivial character.