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The motivic Satake equivalence using perverse Nori motives

Khoa Bang Pham

TL;DR

This work constructs a comprehensive motivic refinement of the geometric Satake equivalence by developing stratified perverse Nori motives and their equivariant and ind-scheme extensions. It establishes a neutral Tannakian Satake category on the affine Grassmannian whose dual is the product of the Langlands dual and the motivic Galois group, thereby capturing rich motivic symmetries beyond the Tate model. The framework supports a robust six-functor formalism, stratified intersection motives, and a convolution structure that mirrors classical Satake while accommodating motivic local systems. The results pave the way for deeper connections between motivic fundamental groups, Tannakian duals, and Beilinson–Drinfeld fusion, with potential applications to motivic Langlands and beyond.

Abstract

In this article, we develop the theory of stratified perverse Nori motives to prove a refinement of the geometric Satake equivalence of Mirković-Vilonen, for which we call the Nori motivic Satake equivalence, in constrast to the "Tate motivic" Satake equivalence of Richarz-Scholbach.

The motivic Satake equivalence using perverse Nori motives

TL;DR

This work constructs a comprehensive motivic refinement of the geometric Satake equivalence by developing stratified perverse Nori motives and their equivariant and ind-scheme extensions. It establishes a neutral Tannakian Satake category on the affine Grassmannian whose dual is the product of the Langlands dual and the motivic Galois group, thereby capturing rich motivic symmetries beyond the Tate model. The framework supports a robust six-functor formalism, stratified intersection motives, and a convolution structure that mirrors classical Satake while accommodating motivic local systems. The results pave the way for deeper connections between motivic fundamental groups, Tannakian duals, and Beilinson–Drinfeld fusion, with potential applications to motivic Langlands and beyond.

Abstract

In this article, we develop the theory of stratified perverse Nori motives to prove a refinement of the geometric Satake equivalence of Mirković-Vilonen, for which we call the Nori motivic Satake equivalence, in constrast to the "Tate motivic" Satake equivalence of Richarz-Scholbach.
Paper Structure (36 sections, 72 theorems, 137 equations)

This paper contains 36 sections, 72 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. The (geometric) Satake equivalence
  3. Why perverse Nori motives?
  4. Formulation of main results
  5. Some results and questions independent of interests
  6. Structure of the manuscript
  7. Perverse Nori Motives
  8. Motivic Perverse Sheaves (d'apres Ivorra-Morel-Terenzi)
  9. Motivic Local Systems
  10. Smooth Descent
  11. Derived Nori Motives on Ind-Schemes
  12. Reminder on ind-schemes and their stratifications
  13. General properties of derived Nori motives on ind-schemes
  14. Stratified derived Nori Motives
  15. Stratified perverse Nori motives
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $\iota \colon X^+ = \coprod_{w \in W} X_w \longrightarrow X$ be a stratified ind-varieties over $k$. Assume that each stratum $X_w$ is smooth, there is a category of stratified derived Nori motives $\mathbf{DN}^b(X,X^+) \subset \mathbf{DN}^b(X)$ together with a Betti realization functor where the right hand side is the derived category with cohomologies being local systems along strata. Moreo

Theorems & Definitions (165)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • ...and 155 more