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A Coherent Version of Geometric Satake Equivalence for Type A

Shiyixin Liang

TL;DR

This work establishes a coherent version of the geometric Satake equivalence for type $A$ by isolating a finite-length, neutral Tannakian subcategory $\mathrm{KP}_0$ inside the abelianized categorified Coulomb branch and proving it is equivalent to $\mathrm{Rep}(\check{G})$. The strategy combines the Cautis–Williams abelian framework with Hodge-module techniques, using the functor $\Xi$ to relate coherent IC-extensions on the Beilinson–Drinfeld Grassmannian to those on the affine Grassmannian, and a fiber functor built from the cohomology of Gr and Hodge-theoretic graded de Rham theory. A key technical feat is the $K$-group computation that matches the convolution product with the tensor structure in $\mathrm{Rep}(\check{G})$, together with the construction of renormalized $r$-matrices to impose a commutativity constraint and yield a Tannakian category. The approach hinges on type $A$-specific features of Hodge modules and symplectic slices (via Xin’s inequalities) and points to possible extensions to other types by adjusting perversities or perverse coherent t-structures. Overall, the results provide a coherent, representation-theoretic interpretation of the Coulomb-branch geometry in the local geometric Langlands setting and suggest a path to broader generalizations.

Abstract

In this paper we prove a coherent version of geometric Satake equivalence proposed in Cautis-Williams' work arXiv:2306.03023 for type A. In their work, they studied an abelian version of the classical limit Satake category, namely, the Koszul perverse heart of the categorified Coulomb branch for adjoint representations. In this paper we study a subcategory generated by a collection of simple objects. We endow this subcategory with a neutral Tannakian structure and identify it with the finite dimensional representation category $\mathrm{Rep}({\check{G}})$ for the Langlands dual group ${\check{G}}$. Our method uses tools in Cautis-Williams theory and a Hodge module description of the coherent IC extensions of differential sheaves in Xin's work arXiv:2503.14890.

A Coherent Version of Geometric Satake Equivalence for Type A

TL;DR

This work establishes a coherent version of the geometric Satake equivalence for type by isolating a finite-length, neutral Tannakian subcategory inside the abelianized categorified Coulomb branch and proving it is equivalent to . The strategy combines the Cautis–Williams abelian framework with Hodge-module techniques, using the functor to relate coherent IC-extensions on the Beilinson–Drinfeld Grassmannian to those on the affine Grassmannian, and a fiber functor built from the cohomology of Gr and Hodge-theoretic graded de Rham theory. A key technical feat is the -group computation that matches the convolution product with the tensor structure in , together with the construction of renormalized -matrices to impose a commutativity constraint and yield a Tannakian category. The approach hinges on type -specific features of Hodge modules and symplectic slices (via Xin’s inequalities) and points to possible extensions to other types by adjusting perversities or perverse coherent t-structures. Overall, the results provide a coherent, representation-theoretic interpretation of the Coulomb-branch geometry in the local geometric Langlands setting and suggest a path to broader generalizations.

Abstract

In this paper we prove a coherent version of geometric Satake equivalence proposed in Cautis-Williams' work arXiv:2306.03023 for type A. In their work, they studied an abelian version of the classical limit Satake category, namely, the Koszul perverse heart of the categorified Coulomb branch for adjoint representations. In this paper we study a subcategory generated by a collection of simple objects. We endow this subcategory with a neutral Tannakian structure and identify it with the finite dimensional representation category for the Langlands dual group . Our method uses tools in Cautis-Williams theory and a Hodge module description of the coherent IC extensions of differential sheaves in Xin's work arXiv:2503.14890.
Paper Structure (60 sections, 46 theorems, 258 equations, 1 figure)

This paper contains 60 sections, 46 theorems, 258 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the classical limit problems
  3. An abelian version by Cautis-Williams
  4. Main theorem
  5. The strategy of proof
  6. Hodge module theory
  7. Further direction
  8. Structure of the paper
  9. Notations and conventions
  10. Acknowledgements
  11. Algebro-geometric background in Cautis-Williams theory
  12. Ind-geometric derived stacks
  13. DG algebras
  14. Derived stacks
  15. Some conventions about morphisms
  16. ...and 45 more sections

Key Result

Theorem 1.3.1

Assume $G$ is of type A. Then the category $\mathrm{KP}_0$ admits a neutral Tannakian structure (in the sense of DM82), and is equivalent to $\mathrm{Rep}({\check{G}})$ as neutral Tannakian categories over $\mathbb{C}$.

Figures (1)

  • Figure 1: Notation used in the paper

Theorems & Definitions (111)

  • Conjecture 1.1.1: BFM05
  • Conjecture 1.2.1: CW23
  • Conjecture 1.2.2: CW23
  • Theorem 1.3.1: Theorem \ref{['main thm']}
  • Theorem 1.4.1
  • Theorem 1.4.2: Corollary \ref{['cor:fiber functor']}, Theorem \ref{['thm: fibre functor']}
  • Proposition 1.4.1: Corollary \ref{['cor: Hodge-de Rham desciption for IC lav']}
  • Theorem 1.4.3: Xin25
  • Conjecture 1.5.1
  • Example 2.1.1
  • ...and 101 more