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Derived geometric Satake equivalence on the Beilinson-Drinfeld Grassmannian with one leg in mixed characteristic

Katsuyuki Bando

Abstract

Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the derived version of this equivalence with one leg. Namely, we show that the derived category of etale sheaves on the local Hecke stack is equivalent to the category of L-group-equivariant perfect complexes over the symmetric algebra of the shifted and (-1)-Tate-twisted Lie algebra.

Derived geometric Satake equivalence on the Beilinson-Drinfeld Grassmannian with one leg in mixed characteristic

Abstract

Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the derived version of this equivalence with one leg. Namely, we show that the derived category of etale sheaves on the local Hecke stack is equivalent to the category of L-group-equivariant perfect complexes over the symmetric algebra of the shifted and (-1)-Tate-twisted Lie algebra.
Paper Structure (34 sections, 67 theorems, 349 equations)

This paper contains 34 sections, 67 theorems, 349 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Fundamental notation
  4. Categories of Representations
  5. Dual group and its Weil action
  6. Notation on diamonds and v-stacks
  7. Notation and preliminaries on geometric Satake
  8. Hecke stacks
  9. Schubert varieties
  10. ULA sheaves and Satake category
  11. Witt vector and equal characteristic Hecke stacks
  12. Cohomology of Hecke stack
  13. Equivalence between categories of ULA sheaves
  14. Cohomology ring of Hecke stack
  15. Weil action on cohomology ring of Hecke stack
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $G$ be a reductive group over $F$. There exists an equivalence of stable monoidal $\infty$-categories, where $\widehat{G}\rtimes W_{F}$ acts on $\widehat{\mathfrak{g}}$ via the adjoint action, the symbol $(-1)$ denotes the $(-1)$-Tate twist, and we regard $\rm{Sym}(\widehat{\mathfrak{g}}(-1)[-2]))$ as a formal dg-algebra. Moreover, it is compatible with the (no

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 114 more