A Coherent Version of Geometric Satake Equivalence for Type A

TL;DR

This paper establishes a coherent version of the geometric Satake equivalence for type A, connecting a subcategory of the Coulomb branch to the representation category of the Langlands dual group using Hodge modules.

Contribution

It introduces a new subcategory with a Tannakian structure and identifies it with the dual group's representation category, extending geometric Satake equivalence.

Findings

Identified a subcategory with Tannakian structure

Established equivalence with the dual group's representation category

Utilized Hodge module techniques in the proof

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.