Harder-Narasimhan strata and $p$-adic period domains

TL;DR

This paper explores the structure of $p$-adic flag varieties using the theory of $G$-bundles, comparing stratifications, and extending results to broader cases with applications to Shimura varieties.

Contribution

It introduces new comparisons between Harder-Narasimhan and Newton strata, and generalizes results to arbitrary cocharacters in $p$-adic geometry.

Findings

Established equivalences in $p$-adic Hodge theory conditions.

Extended stratification results to arbitrary cocharacters.

Applied constructions to Shimura varieties and local analogues.

Abstract

We revisit the Harder-Narasimhan stratification on a minuscule $p$-adic flag variety, by the theory of modifications of $G$-bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of $p$-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated $B_{dR}^+$-affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give applications to $p$-adic geometry of Shimura varieties and their local analogues.