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Coherent sheaves on the stack of Langlands parameters

Xinwen Zhu

TL;DR

The paper proposes a program to realize the arithmetic local and global Langlands correspondences via coherent sheaves on stacks of Langlands parameters, introducing derived moduli spaces and stacks as foundational objects.It develops a detailed toolkit—derived representation spaces, continuous/strongly continuous variants, and local/global parameter stacks—together with conjectural endomorphism algebras and dualities that mirror geometric Langlands intuition in an arithmetic setting.Key contributions include the derived Satake framework, spectral parabolic induction, coherent Springer sheaves, and a broad Conjectural arithmetic LLC, with concrete evidence in local function-field cases and links to patched-module theory and Shimura-variety cohomology.The framework aims to unify local and global aspects, accommodate inner forms and tori, and produce a versatile, functorial mechanism for translating automorphic representations into coherent sheaves on parameter stacks, with potential applications to cohomology of Shtukas and cohomological realizations of Jacquet–Langlands transfers.

Abstract

We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.

Coherent sheaves on the stack of Langlands parameters

TL;DR

The paper proposes a program to realize the arithmetic local and global Langlands correspondences via coherent sheaves on stacks of Langlands parameters, introducing derived moduli spaces and stacks as foundational objects.It develops a detailed toolkit—derived representation spaces, continuous/strongly continuous variants, and local/global parameter stacks—together with conjectural endomorphism algebras and dualities that mirror geometric Langlands intuition in an arithmetic setting.Key contributions include the derived Satake framework, spectral parabolic induction, coherent Springer sheaves, and a broad Conjectural arithmetic LLC, with concrete evidence in local function-field cases and links to patched-module theory and Shimura-variety cohomology.The framework aims to unify local and global aspects, accommodate inner forms and tori, and produce a versatile, functorial mechanism for translating automorphic representations into coherent sheaves on parameter stacks, with potential applications to cohomology of Shtukas and cohomological realizations of Jacquet–Langlands transfers.

Abstract

We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.

Paper Structure

This paper contains 19 sections, 51 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Representation space
  3. The derived category of monoids
  4. The derived representation space
  5. Some examples
  6. Continuous representations
  7. The stack of arithmetic Langlands parameters
  8. The stack of local Langlands parameters
  9. Duality for Tori and symmetries of ${\operatorname{Coh}}({\operatorname{Loc}}_{{}^cG,F,\iota})$
  10. Spectral parabolic induction
  11. The stack of global Langlands parameters
  12. Coherent sheaves on the stack of Langlands parameters
  13. The category of representations of $G(F)$
  14. The groupoids ${\mathbf{W}}_G$ and ${\mathbf{TS}}_G$
  15. Derived Satake isomorphism
  16. ...and 4 more sections

Key Result

Lemma 2.3

There is a canonical equivalence ${\mathbf{Ani}}({\mathbf{Mon}})\cong {\mathbf{Mon}}(\mathbf{Spc})$.

Theorems & Definitions (195)

  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 185 more