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

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.