The reduction of $G$-ordinary crystalline representations with $G$-structure

TL;DR

This paper explores how lattices in $G$-ordinary crystalline representations reduce to crystals with $G$-structure, extending Fargues' filtration to Kisin modules to describe this process.

Contribution

It generalizes Fargues' Harder-Narasimhan filtration to Kisin modules and describes the reduction of $G$-ordinary crystalline representations using $G$-structure.

Findings

Provides a description of lattice reduction in $G$-ordinary crystalline representations.

Extends Fargues' filtration to Kisin modules.

Connects reduction process with $G$-structure in crystalline context.

Abstract

Fontaine's $D_{\mathrm{cris}}$ functor allow us to associate an isocrystal to any crystalline representation. For a reductive group $G$, we study the reduction of lattices inside a germ of crystalline representations with $G$-structure $V$ to lattices (which are crystals) with $G$-structure inside $D_{\mathrm{cris}}(V)$. Using Kisin modules theory, we give a description of this reduction in terms of $G$, in the case when the representation $V$ is ($G$-)ordinary. In order to do that, first we need to generalize Fargues' construction of the Harder-Narasimhan filtration for $p$-divisible groups to Kisin modules.