The geometric Satake equivalence for integral motives

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.

Final version, to appear in Compositio Mathematica

References in corpus (12)

• Triangulated categories of mixed motives
• The six operations in equivariant motivic homotopy theory
• Parity Sheaves
• A Satake isomorphism in characteristic p
• A comparison of motivic and classical homotopy theories
• The intersection motive of the moduli stack of shtukas
• The Test Function Conjecture for Local Models of Weil-restricted groups
• Intermediate extension of Chow motives of Abelian type
• Tannakian duality over Dedekind rings and applications
• Perverse motives and graded derived category $\mathcal{O}$
• The motivic Satake equivalence
• Künneth formulas for motives and additivity of traces

Cited by in corpus (1)

• Geometrization of the Satake transform for mod $p$ Hecke algebras