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On the weak Harder-Narasimhan stratification on $B_{\mathrm{dR}}^+$-affine Grassmannian

Miaofen Chen, Jilong Tong

Abstract

We consider the Harder-Narasimhan formalism on the category of normed isocrystals and show that the Harder-Narasimhan filtration is compatible with tensor products which generalizes a result of Cornut. As an application of this result, we are able to define a (weak) Harder-Narasimhan stratification on the $B_{\mathrm{dR}}^+$-affine Grassmannian for arbitrary $(G, b, μ)$. When $μ$ is minuscule, it corresponds to the Harder-Narasimhan stratification on the flag varieties defined by Dat-Orlik-Rapoport. And when $b$ is basic, it's studied by Nguyen-Viehmann and Shen. We study the basic geometric properties of the Harder-Narasimhan stratification, such as non-emptiness, dimension and its relation with other stratifications.

On the weak Harder-Narasimhan stratification on $B_{\mathrm{dR}}^+$-affine Grassmannian

Abstract

We consider the Harder-Narasimhan formalism on the category of normed isocrystals and show that the Harder-Narasimhan filtration is compatible with tensor products which generalizes a result of Cornut. As an application of this result, we are able to define a (weak) Harder-Narasimhan stratification on the -affine Grassmannian for arbitrary . When is minuscule, it corresponds to the Harder-Narasimhan stratification on the flag varieties defined by Dat-Orlik-Rapoport. And when is basic, it's studied by Nguyen-Viehmann and Shen. We study the basic geometric properties of the Harder-Narasimhan stratification, such as non-emptiness, dimension and its relation with other stratifications.
Paper Structure (32 sections, 41 theorems, 334 equations)

This paper contains 32 sections, 41 theorems, 334 equations.

Table of Contents

  1. preliminaries on $G$-bundles on the Fargues-Fontaine curve
  2. $G$-bundles on the Fargues-Fontaine curve
  3. Fargues-Fontaine curve
  4. The Kottwitz set
  5. $G$-bundles
  6. Harder-Narasimhan reduction in the quasi-split case
  7. $B_{\mathrm{dR}}^+$-affine Grassmannian
  8. Schubert varieties
  9. Generalized semi-infinite orbits
  10. $B_{\mathrm{dR}}^+$-affine Grassmannian and modifications of $G$-bundles
  11. Harder-Narasimhan filtration of normed isocrystals
  12. The category $\mathbf{NIsoc}_{\breve F|F}^{K}$
  13. Harder-Narasimhan filtration for normed isocrystals
  14. Compatibility of the Harder-Narasimhan filtration with tensor products
  15. The vectorial Tits building for $\mathbf{GL}_n$
  16. ...and 17 more sections

Key Result

Theorem 1

The functor $\mathcal{F}_{\rm HN}: \mathbf{BunIsoc}_{\breve F|F}^{B_{\mathrm{dR}}}\rightarrow \mathbf{F}(\mathbf{Isoc}_{\breve F|F})$ is compatible with tensor products.

Theorems & Definitions (107)

  • Theorem : Theorem \ref{['thm_compatible tensor']}
  • Remark 1.1
  • Lemma 1.2
  • Proposition 1.3: Sch
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6: FS
  • Remark 1.7
  • Lemma 1.8
  • proof
  • ...and 97 more