On Newton strata in the $B_{dR}^+$-Grassmannian

TL;DR

This paper investigates the structure of Newton strata and parabolic reductions of G-bundles on the Fargues-Fontaine curve, establishing closure relations and confirming a conjecture about intersections with the weakly admissible locus in the $B_{dR}^+$-Grassmannian.

Contribution

It proves that the closure relations on $|Bun_G|$ match the opposite of the partial order on B(G) and confirms Chen's conjecture on intersections of Newton strata with the weakly admissible locus.

Findings

Closure relations on $|Bun_G|$ match the opposite of B(G) order.

Every non-Hodge-Newton decomposable Newton stratum intersects the weakly admissible locus.

Identifies classical points in Newton strata.

Abstract

We study parabolic reductions and Newton points of G-bundles on the Fargues-Fontaine curve and the Newton stratification on the $B_{dR}^+$-Grassmannian for any reductive group G. Let $Bun_G$ be the stack of G-bundles on the Fargues-Fontaine curve. Our first main result is to show that under the identification of the points of $Bun_G$ with Kottwitz's set B(G), the closure relations on the topological space $|Bun_G|$ coincide with the opposite of the usual partial order on B(G). Furthermore, we prove that every non-Hodge-Newton decomposable Newton stratum in a minuscule affine Schubert cell in the $B_{dR}^+$-Grassmannian intersects the weakly admissible locus, proving a conjecture of Chen. On the way, we study several interesting properties of parabolic reductions of $G$-bundles, and determine which Newton strata have classical points.