Quantization of the universal centralizer and central D-modules

TL;DR

This paper establishes a deep connection between the quantization of universal centralizers in complex reductive groups and categories of D-modules, proving several conjectures and constructing new monoidal equivalences.

Contribution

It constructs a braided monoidal equivalence (Knop-Ng extsuperscript{o} functor) linking universal centralizer quantizations to bi-Whittaker D-modules, extending prior categorical results.

Findings

Proves conjectures of Ben-Zvi and Gunningham relating to parabolic induction.

Establishes a D-module analogue of known categorical equivalences.

Relates the universal centralizer quantization to the spherical nil-DAHA.

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-Ng\^o 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.