Cycles relations in the affine grassmannian and applications to Breuil--Mézard for G-crystalline representations

For a split reductive group $G$ we realise identities in the Grothendieck group of $\widehat{G}$-representation in terms of cycle relations between certain closed subschemes inside the affine grassmannian. These closed subschemes are obtained as a degeneration of $e$-fold products of flag varieties and, under a bound on the Hodge type, we relate the geometry of these degenerations to that of moduli spaces of $G$-valued crystalline representations of $\mathrm{Gal}(\bar{K}\mathrm{/}K)$ for $K\mathrm{/}{\mathbb{Q}}_p$ a finite extension with ramification degree $e$. By transferring the aforementioned cycle relations to these moduli spaces we deduce one direction of the Breuil–Mézard conjecture for $G$-valued crystalline representations with small Hodge type.