Degenerating products of flag varieties and applications to the Breuil--Mezard conjecture

Abstract

We consider closed subschemes in the affine grassmannian obtained by degenerating $e$-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For $G= \operatorname{GL}_2$, and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of $G$. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton--Gee stack classifying $p$-adic representations of the Galois group of a finite extension of $\mathbb{Q}_p$. As an application we prove new cases of the Breuil--M\'ezard conjecture in dimension two.