Classification of nondegenerate $G$-categories (with an appendix written jointly with Germ\'an Stefanich)

TL;DR

This paper classifies a specific class of categories with reductive group actions using root data, and applies these results to enhance equivalences in geometric representation theory.

Contribution

It provides a complete classification of nondegenerate G-categories via root data and strengthens existing equivalences in the theory of bi-Whittaker D-modules.

Findings

Classifies nondegenerate G-categories using root data.

Upgrades Ginzburg-Lonergan equivalence to a monoidal one.

Shows parabolic restriction sheaves acquire Weyl group symmetry.

Abstract

We classify a "dense open" subset of categories with an action of a reductive group, which we call nondegenerate categories, entirely in terms of the root datum of the group. As an application of our methods, we also: (1) Upgrade an equivalence of Ginzburg and Lonergan, which identifies the category of bi-Whittaker $\mathcal{D}$-modules on a reductive group with the category of $\tilde{W}$-equivariant sheaves on a dual Cartan subalgebra $\mathfrak{t}^*$ which descend to the coarse quotient $\mathfrak{t}^*//\tilde{W}$, to a monoidal equivalence (where $\tilde{W}$ denotes the extended affine Weyl group) and (2) Show the parabolic restriction of a very central sheaf acquires a Weyl group equivariant structure such that the associated equivariant sheaf descends to the coarse quotient $\mathfrak{t}^*//\tilde{W}$, providing evidence for a conjecture of Ben-Zvi-Gunningham on parabolic restriction.