The motivic Satake equivalence

TL;DR

This paper refines the geometric Satake equivalence by establishing a precise correspondence between mixed Tate motives on a certain double quotient and representations of a modified Langlands dual group, advancing the understanding of geometric Langlands.

Contribution

It introduces a refined equivalence linking mixed Tate motives on the double quotient to representations of Deligne's modified dual group, enhancing the geometric Satake framework.

Findings

Establishes an equivalence between mixed Tate motives and dual group representations.

Provides a new perspective on the geometric Satake correspondence.

Advances the understanding of the Langlands program through motive-based methods.

Abstract

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirkovi\'c-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^+ G \backslash LG / L^+ G$ and representations of Deligne's modification of the Langlands dual group $\hat G$.