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The geometric Satake equivalence for integral motives

Robert Cass, Thibaud van den Hove, Jakob Scholbach

TL;DR

The paper develops the geometric Satake program in the setting of mixed Tate motives with integral coefficients, focusing on Beilinson–Drinfeld Grassmannians and their stratifications. It proves a global integral motivic Satake equivalence Sat^{G,I} ≅ Rep_{\\hat G^{I}}(MTM(S)) under Beilinson–Soulé vanishing, and constructs a fusion product, a fiber functor, and a Tannakian reconstruction that identifies the Satake category with representations of a graded dual group while linking to Vinberg monoids. The approach unifies Satake-type phenomena across several sheaf theories in the integral/motivic context and sets up a robust framework for global Langlands applications, including tamely ramified motivic centers, mod-p aspects, and potential l-independence via motivic foundations. The work lays groundwork for future global Langlands parametrizations over function fields and connects to central functors and motivic analogues of Deligne’s dual group, aiming to advance both geometric representation theory and arithmetic geometry with integral motivic techniques.

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.

The geometric Satake equivalence for integral motives

TL;DR

The paper develops the geometric Satake program in the setting of mixed Tate motives with integral coefficients, focusing on Beilinson–Drinfeld Grassmannians and their stratifications. It proves a global integral motivic Satake equivalence Sat^{G,I} ≅ Rep_{\\hat G^{I}}(MTM(S)) under Beilinson–Soulé vanishing, and constructs a fusion product, a fiber functor, and a Tannakian reconstruction that identifies the Satake category with representations of a graded dual group while linking to Vinberg monoids. The approach unifies Satake-type phenomena across several sheaf theories in the integral/motivic context and sets up a robust framework for global Langlands applications, including tamely ramified motivic centers, mod-p aspects, and potential l-independence via motivic foundations. The work lays groundwork for future global Langlands parametrizations over function fields and connects to central functors and motivic analogues of Deligne’s dual group, aiming to advance both geometric representation theory and arithmetic geometry with integral motivic techniques.

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.
Paper Structure (61 sections, 111 theorems, 214 equations)

This paper contains 61 sections, 111 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Geometry of the Beilinson--Drinfeld affine Grassmannians
  3. Application: a global integral motivic Satake equivalence
  4. Future directions
  5. Outline
  6. Motivic sheaves
  7. Recollections
  8. Motives
  9. Betti realization
  10. Reduced motives
  11. Functoriality
  12. Motives on prestacks
  13. Hyperbolic localization
  14. Stratified Tate motives
  15. t-structures
  16. ...and 46 more sections

Key Result

Theorem 1.1

The stratification of $\mathop{\rm Gr}\nolimits_{G,I}$ in BD.strata, which combines the Schubert cells in $\mathop{\rm Gr}\nolimits_G$ with a stratification of $X^I$, is admissible Whitney--Tate in the sense of Definitions Whitney--Tate and admissible.stratification.

Theorems & Definitions (293)

  • Theorem 1.1: \ref{['BD.WT']}
  • Theorem 1.2: \ref{['cellularity of intersection:torus']}
  • Theorem 1.3: § \ref{['sect--Tannakian']}, \ref{['MTM of local Grassmannian']}
  • Theorem 1.4
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 283 more