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

Abstract

We prove an arithmetic analogue of Kashiwara regularity for filtered strongly equivariant Berthelot arithmetic $\mathscr D^\dagger$-modules on formal flag varieties. Let $\mathcal G$ be a split connected reductive group scheme over a complete discrete valuation ring of mixed characteristic, let $\mathfrak X=\widehat{\mathcal G/\mathcal B}$ be the formal flag variety, and let $\mathcal K\subseteq\mathcal G$ be a smooth closed subgroup whose special fiber acts on $X_s$ with finitely many separable orbits. We introduce a filtered strong equivariance condition requiring infinitesimal equivariance to be realized on good finite-level models. This condition allows the principal symbols of the fundamental vector fields to be controlled at the level where Berthelot characteristic varieties are defined. We prove that the characteristic variety of every filtered strongly equivariant coherent $\mathscr D^\dagger_{\mathfrak X,\mathbb Q}$-module is contained in the union of conormal bundles to the $\mathcal K_s$-orbits; hence such modules are holonomic. In the Frobenius range, Caro's stability theorem for $F$-holonomicity over smooth projective formal schemes upgrades this to geometric overholonomicity. As a consequence, simple Frobenius filtered strongly equivariant coherent arithmetic $\mathscr D^\dagger$-modules are classified by pairs $(O,E)$, where $O$ is a $\mathcal K_s$-orbit and $E$ is an irreducible $\mathcal K$-equivariant overconvergent $F$-isocrystal on $(O,\overline O)$ whose intermediate extension satisfies the filtered strong condition. The result is a regularity theorem under a finite-level equivariance hypothesis, rather than an existence theorem for that hypothesis.