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For a split reductive group $G$ we realise identities in the Grothendieckgroup of $\widehat{G}$-representation in terms of cycle relations betweencertain closed subschemes inside the affine grassmannian. These closedsubschemes are obtained as a degeneration of $e$-fold products of flagvarieties and, under a bound on the Hodge type, we relate the geometry of thesedegenerations to that of moduli spaces of $G$-valued crystallinerepresentations of $\operatorname{Gal}(\overline{K}/K)$ for $K/\mathbb{Q}_p$ afinite extension with ramification degree $e$. By transferring theaforementioned cycle relations to these moduli spaces we deduce one directionof the Breuil--M\'ezard conjecture for $G$-valued crystalline representationswith small Hodge type.

Cycles relations in the affine grassmannian and applications to Breuil--M\'ezard for G-crystalline representations