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Exponential motives on the affine Grassmannian

Robert Cass, Thibaud van den Hove, Jakob Scholbach

Abstract

We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on affine flag varieties concerning Tate motives and t-structures. We use this to prove a motivic Casselman-Shalika equivalence, relating exponential Tate motives on the affine Grassmannian to ind-coherent sheaves on the classifying stack of the Langlands dual group. The decategorification of this equivalence provides a new construction of the Whittaker module for the spherical Hecke algebra which works for arbitrary coefficients, including a generic version.

Exponential motives on the affine Grassmannian

Abstract

We develop a notion of exponential motives on general prestacks equipped with a -action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on affine flag varieties concerning Tate motives and t-structures. We use this to prove a motivic Casselman-Shalika equivalence, relating exponential Tate motives on the affine Grassmannian to ind-coherent sheaves on the classifying stack of the Langlands dual group. The decategorification of this equivalence provides a new construction of the Whittaker module for the spherical Hecke algebra which works for arbitrary coefficients, including a generic version.
Paper Structure (46 sections, 102 theorems, 300 equations)

This paper contains 46 sections, 102 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Exponential motives and Whittaker motives
  3. Exponential geometry of affine flag varieties
  4. A motivic Casselman--Shalika equivalence
  5. The generic Whittaker module
  6. Exponential motives
  7. Preliminaries and recollections
  8. Motives
  9. Motives on prestacks; equivariant motives
  10. Complements on mixed Tate motives
  11. Local and colocal objects; quotient categories
  12. Definitions of exponential motives and basic examples
  13. Functoriality for exponential motives
  14. Exponential Tate motives
  15. Whittaker motives
  16. ...and 31 more sections

Key Result

Theorem 1.2

For any prestack $X / {{\mathbf F}_p}$ acted upon by $\mathbf {G}_\mathrm a \rtimes \mathbf {G}_\mathrm m$, there is an equivalence between the category of $\mathbf {G}_\mathrm m$-equivariant exponential étale motives (defined in intro:eqexp) and the category of Whittaker motives.

Theorems & Definitions (244)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Example 2.2
  • ...and 234 more