Satake equivalence for Hodge modules on affine Grassmannians

TL;DR

This paper establishes a Satake equivalence for Hodge modules on affine Grassmannians, connecting geometric representation theory with Hodge theory and the Langlands dual group.

Contribution

It constructs a Tannakian category structure on G_O-equivariant pure Hodge modules and proves its equivalence to a twisted tensor product involving the Langlands dual group.

Findings

Category of G_O-equivariant pure Hodge modules is a neutral Tannakian category.

Equivalence to a twisted tensor product of dual group representations and Hodge structures.

Provides a new geometric realization linking Hodge modules and Langlands duality.

Abstract

For a reductive group $G$ we equip the category of $G_\mathcal{O}$-equivariant polarizable pure Hodge modules on the affine Grassmannian $\mathrm{Gr}_G$ with a structure of neutral Tannakian category. We show that it is equivalent to a twisted tensor product of the category of representations of the Langlands dual group and the category of pure polarizable Hodge structures.