Quantum category O vs affine Hecke category

TL;DR

This paper establishes an equivalence between the quantum category O at an odd root of unity and the affine Hecke category by deforming categories and constructing functors, revealing deep structural connections.

Contribution

It introduces a new approach to relate quantum category O and affine Hecke category through deformation theory and functor construction, including the development of Rouquier-Soergel theory.

Findings

Proved equivalence of highest weight categories between quantum O and affine Hecke categories.

Constructed functors from deformed categories to bimodule categories that are full embeddings.

Developed Rouquier-Soergel theory to analyze standardly filtered objects.

Abstract

The goal of this paper is to relate the quantum category $\mathcal{O}$ (known also as the category of modules over the mixed quantum group) at an odd root of unity to the affine Hecke category. Namely, we prove equivalences of highest weight categories between integral blocks of the affine category $\mathcal{O}$ and the heart of the so called ``new'' t-structure on the affine Hecke category. In order to do this we deform our categories over the formal neighborhood of $0$ in the dual affine Cartan and show that the categories of standardly filtered objects in the deformations are equivalent. For this, we construct functors from the deformed categories to the category of bimodules over the formal power series on the affine Cartan. Then we use what we call the Rouquier-Soergel theory, also developed in this paper, to show that on the categories of standardly filtered objects, these functors are full embeddings with the same image.