On nonemptiness of Newton strata in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$

TL;DR

This paper classifies all nonempty Newton strata within minuscule Schubert cells of the $B_ ext{dR}^+$-Grassmannian for $ ext{GL}_n$, providing explicit conditions for many cases and using induction on n with vector bundle extensions.

Contribution

It offers a comprehensive classification of nonempty Newton strata in the $B_ ext{dR}^+$-Grassmannian for $ ext{GL}_n$, extending previous results and providing explicit criteria.

Findings

Classified all nonempty Newton strata in minuscule Schubert cells.

Provided explicit conditions for nonemptiness based on Newton polygons.

Used induction on n and vector bundle extension classification on the Fargues-Fontaine curve.

Abstract

We study the Newton stratification in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$ associated to an arbitrary (possibly nonbasic) element of $B(\mathrm{GL}_n)$. Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in $B(\mathrm{GL}_n)$, our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues-Fontaine curve.