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

Robin Bartlett

Abstract

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 $\operatorname{Gal}(\overline{K}/K)$ for $K/\mathbb{Q}_p$ 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.

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

Abstract

For a split reductive group we realise identities in the Grothendieck group of -representation in terms of cycle relations between certain closed subschemes inside the affine grassmannian. These closed subschemes are obtained as a degeneration of -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 -valued crystalline representations of for a finite extension with ramification degree . By transferring the aforementioned cycle relations to these moduli spaces we deduce one direction of the Breuil--Mézard conjecture for -valued crystalline representations with small Hodge type.
Paper Structure (22 sections, 45 theorems, 230 equations)

This paper contains 22 sections, 45 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Cycles identities in the affine grassmannian
  3. Notation
  4. Torsors
  5. Affine grassmannians
  6. Various Schubert varieties
  7. Approximations via $\operatorname{Gr}_G^\nabla$
  8. Computations in $\operatorname{Gr}_G^\nabla$
  9. Naive cycle identities
  10. Irreducibility
  11. Equivariant sheaves and their cycles
  12. Determinant line bundles
  13. Main theorem
  14. Some representation theory
  15. Cycle identities in moduli spaces of crystalline representations
  16. ...and 7 more sections

Key Result

Theorem 1.1

Assume $G$ admits a twisting element $\rho$, i.e. a cocharacter pairing to $1$ with all simple roots of $G$. Then, for any tuple $\mu = (\mu_1,\ldots,\mu_e)$ of strictly dominant cocharacters of $G$ (i.e. $\mu_i-\rho$ is dominant) satisfying for all roots $\alpha^\vee$ of $G$ (when $\operatorname{char}\mathbb{F} =0$ this condition is not needed) one has identities of $e\operatorname{dim}G/B$-dime

Theorems & Definitions (109)

  • Theorem 1.1
  • Theorem 1.3
  • Definition 2.2
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Corollary 4.5
  • Lemma 4.8
  • proof
  • ...and 99 more