The geometric Satake equivalence for integral motives

TL;DR

This paper establishes a refined geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum, advancing the understanding of motives and their symmetries.

Contribution

It introduces new geometric results like Whitney--Tate stratifications and cellular decompositions, and constructs integral versions of dual groups and monoids, extending prior work.

Findings

Proved geometric Satake equivalence for mixed Tate motives over integers

Developed Whitney--Tate stratifications of Beilinson--Drinfeld Grassmannians

Constructed integral dual groups and Vinberg's monoid

Abstract

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney--Tate stratifications of Beilinson--Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne's modification of the dual group and a modified form of Vinberg's monoid over the integers.