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Central motives on parahoric flag varieties

Robert Cass, Thibaud van den Hove, Jakob Scholbach

TL;DR

The paper constructs a motivic refinement of Gaitsgory’s central functor for integral motivic sheaves, ensuring preservation of stratified Tate motives and a t-exact central functor via Wakimoto filtrations. It develops a motivic unipotent nearby cycles formalism, extends Beilinson–Drinfeld BD geometry to the motivic setting, and proves both monoidality and centrality properties that connect to the motivic Satake equivalence. A key outcome is a motivic decategorification that yields a center isomorphism between generic spherical and parahoric Hecke algebras, advancing a unified, ℓ-free approach to Langlands-type correspondences. The framework integrates higher algebra, six-functor formalism, and hyperbolic localization to establish t-exactness and full compatibility across Beilinson–Drinfeld geometry, central functors, and generic Hecke algebras, with implications for a motivic Arkhipov–Bezrukavnikov program.

Abstract

We construct a refinement of Gaitsgory's central functor for integral motivic sheaves, and show it preserves stratified Tate motives. Towards this end, we develop a reformulation of unipotent motivic nearby cycles, which also works over higher-dimensional bases. We moreover introduce Wakimoto motives and use them to show that our motivic central functor is t-exact. A decategorification of these functors yields a new approach to generic Hecke algebras for general parahorics.

Central motives on parahoric flag varieties

TL;DR

The paper constructs a motivic refinement of Gaitsgory’s central functor for integral motivic sheaves, ensuring preservation of stratified Tate motives and a t-exact central functor via Wakimoto filtrations. It develops a motivic unipotent nearby cycles formalism, extends Beilinson–Drinfeld BD geometry to the motivic setting, and proves both monoidality and centrality properties that connect to the motivic Satake equivalence. A key outcome is a motivic decategorification that yields a center isomorphism between generic spherical and parahoric Hecke algebras, advancing a unified, ℓ-free approach to Langlands-type correspondences. The framework integrates higher algebra, six-functor formalism, and hyperbolic localization to establish t-exactness and full compatibility across Beilinson–Drinfeld geometry, central functors, and generic Hecke algebras, with implications for a motivic Arkhipov–Bezrukavnikov program.

Abstract

We construct a refinement of Gaitsgory's central functor for integral motivic sheaves, and show it preserves stratified Tate motives. Towards this end, we develop a reformulation of unipotent motivic nearby cycles, which also works over higher-dimensional bases. We moreover introduce Wakimoto motives and use them to show that our motivic central functor is t-exact. A decategorification of these functors yields a new approach to generic Hecke algebras for general parahorics.
Paper Structure (67 sections, 83 theorems, 226 equations)

This paper contains 67 sections, 83 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Geometry of Beilinson--Drinfeld Grassmannians
  3. Revisiting unipotent nearby cycles
  4. The central functor
  5. Wakimoto motives
  6. Generic Hecke algebras
  7. Preliminaries
  8. Higher algebra
  9. Monoidality and adjoints
  10. Koszul duality
  11. Motives
  12. Six functors
  13. Stratified Tate motives
  14. Reduced motives
  15. Mixed Tate motives
  16. ...and 52 more sections

Key Result

Theorem 1.1

The above stratifications of $(\mathop{\rm Gr}\nolimits_{\mathcal{G}_{\mathbf{f}}})_\eta$ and $(\mathop{\rm Gr}\nolimits_{\mathcal{G}_\mathbf{f}})_{s}$ determine a universally anti-effective Whitney--Tate stratification of $\mathop{\rm Gr}\nolimits_{\mathcal{G}_{\mathbf{f}}}$, in the sense of Cassvd

Theorems & Definitions (221)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 211 more