Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks: Koszul duality

TL;DR

This paper establishes a deep categorical equivalence between filtered $$-modules on smooth stacks and $S^1$-equivariant ind-coherent sheaves on their formal loop spaces, with implications for geometric Langlands.

Contribution

It introduces a novel equivalence linking filtered $$-modules and equivariant sheaves on loop spaces, advancing the understanding of categorical structures in geometric representation theory.

Findings

Equivalence between filtered $$-modules and $S^1$-equivariant ind-coherent sheaves.

Identification of special and generic fibers with coherent sheaves and constructible sheaves.

Application to categorical traces and local Langlands correspondences.

Abstract

Motivated by applications to the categorical and geometric local Langlands correspondences, we establish an equivalence between the category of filtered $\mathcal{D}$-modules on a smooth stack $X$ and the category of $S^1$-equivariant ind-coherent sheaves on its formal loop space $\widehat{\mathcal{L}} X$, exchanging compact $\mathcal{D}$-modules with coherent sheaves, and coherent $\mathcal{D}$-modules with continuous ind-coherent sheaves. The equivalence yields a sheaf of categories over $\mathbb{A}^1/\mathbb{G}_m$ whose special fiber is a category of coherent sheaves on stacks appearing in categorical traces, and whose generic fiber is a category of equivariant constructible sheaves.