Singular localization for Quantum groups at generic $q$

TL;DR

This paper develops a framework for quantizing parabolic flag manifolds and establishes a singular version of the Beilinson-Bernstein localization theorem for quantum groups at generic q, extending the understanding of quantum D-modules.

Contribution

It introduces a quantization of parabolic flag manifolds and proves a singular Beilinson-Bernstein localization theorem for generic q in quantum groups.

Findings

Computed global sections for all q in C*

Established a singular Beilinson-Bernstein localization theorem

Described categories of equivariant quantum D-modules

Abstract

We quantize parabolic flag manifolds and describe categories of equivariant quantum $\D$-modules on them at a singular central character. We compute global sections at any $q \in \C^*$ and we also prove a singular version of Beilinson-Bernstein localization for a quantized enveloping algebra $\Uq(\g)$, when $q$ is generic.