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

2023/05/10 by Robin Bartlett, Bartlett, Robin
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Advanced Mathematical Identities

paper · doi:10.48550/arxiv.2305.06455

Abstract

For a split reductive group G we realise identities in the Grothendieck group of \widehatG-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 Gal(K/K) for K/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.

Related

• Degenerating products of flag varieties and applications to the Breuil--Mezard conjecture
• Mirror symmetry and the Breuil-Mézard Conjecture
• Crystalline representations of G_Qp^a with coefficients
• Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties
• Irreducibility of some crystalline loci with irregular Hodge--Tate weights