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Quantization of the universal centralizer and central D-modules

Tom Gannon, Victor Ginzburg

TL;DR

This work develops a D-module quantitative framework for the universal centralizer by quantizing its Hopf-algebroid structure via the spherical nil-DAHA and establishing a braided monoidal Knop–Ngô functor from 𝔍-modules to the center of Harish-Chandra bimodules. It connects the 𝔍-module category with very central D-modules on G through parabolic induction, Kostant–Whittaker reduction, and the Miura bimodule, yielding exact vanishing results and transfer functors that realize Braverman–Kazhdan gamma-sheaf phenomena in the D-module setting. The construction passes through a robust Drinfeld-center framework, including a relative center and a Drinfeld-center formalism for Hopf algebroids, enabling a functorial transfer that recovers Ngo’s morphism as a coalgebra map and clarifies the abelian hearts of central modules. The results give precise categorical realizations of Langlands-type functoriality within D-modules, provide parabolic-transfer mechanisms, and furnish perverse-sheaf analogues, advancing the bridge between geometric representation theory and quantum Ngo actions. Overall, the paper delivers a comprehensive quantization pipeline for universal centralizers and derives key exactness/vanishing statements applicable to central D-modules and gamma-sheaves.

Abstract

The group scheme of universal centralizers of a complex reductive group $G$ has a quantization called the spherical nil-DAHA. The category of modules over this ring is equivalent, as a symmetric monoidal category, to the category of bi-Whittaker $D$-modules on $G$. We construct a braided monoidal equivalence, called the Knop-Ngô functor, of this category with a full monoidal subcategory of the abelian category of $\mathrm{Ad}(G)$-equivariant $D$-modules, establishing a $D$-module abelian counterpart of an equivalence established by Bezrukavnikov and Deshpande, in a different way. As an application of our methods, we prove conjectures of Ben-Zvi and Gunningham by relating this equivalence to parabolic induction and prove a conjecture of Braverman and Kazhdan in the $D$-module setting.

Quantization of the universal centralizer and central D-modules

TL;DR

This work develops a D-module quantitative framework for the universal centralizer by quantizing its Hopf-algebroid structure via the spherical nil-DAHA and establishing a braided monoidal Knop–Ngô functor from 𝔍-modules to the center of Harish-Chandra bimodules. It connects the 𝔍-module category with very central D-modules on G through parabolic induction, Kostant–Whittaker reduction, and the Miura bimodule, yielding exact vanishing results and transfer functors that realize Braverman–Kazhdan gamma-sheaf phenomena in the D-module setting. The construction passes through a robust Drinfeld-center framework, including a relative center and a Drinfeld-center formalism for Hopf algebroids, enabling a functorial transfer that recovers Ngo’s morphism as a coalgebra map and clarifies the abelian hearts of central modules. The results give precise categorical realizations of Langlands-type functoriality within D-modules, provide parabolic-transfer mechanisms, and furnish perverse-sheaf analogues, advancing the bridge between geometric representation theory and quantum Ngo actions. Overall, the paper delivers a comprehensive quantization pipeline for universal centralizers and derives key exactness/vanishing statements applicable to central D-modules and gamma-sheaves.

Abstract

The group scheme of universal centralizers of a complex reductive group has a quantization called the spherical nil-DAHA. The category of modules over this ring is equivalent, as a symmetric monoidal category, to the category of bi-Whittaker -modules on . We construct a braided monoidal equivalence, called the Knop-Ngô functor, of this category with a full monoidal subcategory of the abelian category of -equivariant -modules, establishing a -module abelian counterpart of an equivalence established by Bezrukavnikov and Deshpande, in a different way. As an application of our methods, we prove conjectures of Ben-Zvi and Gunningham by relating this equivalence to parabolic induction and prove a conjecture of Braverman and Kazhdan in the -module setting.
Paper Structure (35 sections, 46 theorems, 179 equations)

This paper contains 35 sections, 46 theorems, 179 equations.

Table of Contents

  1. Main results
  2. Knop-Ngô morphism
  3. Parabolic induction and very central $D$-modules
  4. The exactness and vanishing conjectures
  5. Transfer of very central $D$-modules
  6. Outline
  7. Acknowledgements
  8. Monoidal structure on $D(G)$-modules
  9. Conventions
  10. Kostant-Whittaker Reduction
  11. Classical Kostant-Whittaker reduction
  12. Kostant-Whittaker reduction
  13. Bi-Whittaker reduction and $\mathfrak{J}_{\hbar}$-comodules
  14. Kostant-Whittaker reduction and the Miura bimodule
  15. The Miura bimodule and the Gelfand-Graev action
  16. ...and 20 more sections

Key Result

Theorem 1.1.5

${{\enspace}\sf {(i)}}\;$ The functor $\mathfrak{J}\mathop{\mathrm{\text{-}{\mathrm{mod}}}}\nolimits \to {\mathcal{HC}},\, E\mapsto U\mathfrak{g} \otimes_{Z\mathfrak{g}} E$, has a canonical lift to a fully faithful, braided monoidal, exact functor $\Upsilon : {\mathfrak{J}}\mathop{\mathrm{\text{-}{\

Theorems & Definitions (96)

  • Theorem 1.1.5
  • Corollary 1.1.6
  • Remark 1.1.7
  • Remark 1.1.8: Comparison to work of Ben-Zvi and Gunningham
  • Corollary 1.2.2
  • Theorem 1.2.5
  • Remark 1.2.7
  • Theorem 1.3.2
  • Proposition 1.4.2
  • Remark 1.4.3
  • ...and 86 more