Weakly admissible locus and Newton stratification in p-adic Hodge theory

TL;DR

This paper investigates the relationship between the weakly admissible locus and Newton stratification in p-adic Hodge theory, providing conditions for their maximality and criteria for Newton strata containment, with special results for general linear groups.

Contribution

It establishes the equivalence between weakly fully HN-decomposable conditions and the maximality of the weakly admissible locus, extending previous stratification theories in p-adic Hodge theory.

Findings

The weakly admissible locus is a union of Newton strata if and only if the pair (G, μ) is weakly fully HN-decomposable.

Provides a criterion involving G-bundles to determine if a Newton stratum is contained in the weakly admissible locus.

For G=GL(n), offers a combinatorial criterion to check if a vector bundle is an extension of two given bundles.

Abstract

The basic admissible locus $\mathcal{F}(G, \mu, b)^a$ inside the flag variety $\mathcal{F}(G, \mu)$, attached to a reductive group $G$ with a minuscule cocharacter $\mu$ of $G$, is a $p$-adic analogue of the complex analytic period spaces. It has an algebraic approximation $\mathcal{F}(G, \mu, b)^{wa}$ inside the flag variety, called the weakly admissible locus. On the flag variety $\mathcal{F}(G, \mu)$, we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that $\mathcal{F}(G, \mu, b)^{wa}$ is maximal (in the sense that it's a union of Newton strata) is equivalent to $(G, \mu)$ weakly fully HN-decomposable, it's also equivalent to the condition that the Newton stratification is finer than the Harder-Narasimhan stratification. These equivalent conditions are generalizations of the fully HN-decomposable condition and the weakly accessible condition. Moreover, we give a criterion to determine whether a Newton stratum is completely contained in the weakly admissible locus involving $G$-bundles as extensions of $M$-bundles over the Fargues-Fontaine curve, where $M$ is a Levi subgroup of $G$. When $G=\mathrm{GL}_n$, we also give a combinatorial inductive criterion to determine whether a vector bundle over the Fargues-Fontaine curve is an extension of two given vector bundles.