Global quantum differential operators on quantum flag manifolds, theorems of Duflo and Kostant

TL;DR

This paper provides a new proof that the global sections of quantum differential operators on quantum flag manifolds correspond to the quantum group, and derives quantum analogs of classical theorems of Duflo and Kostant.

Contribution

It offers a simplified proof of a key isomorphism and extends classical theorems to the quantum setting, clarifying the structure of quantum groups and their centers.

Findings

Global sections of quantum differential operators equal the quantum group.

Quantum versions of Duflo and Kostant theorems are established.

Description of the center of the ad-integrable quantum group part.

Abstract

We give a short new proof for the theorem that global sections of the sheaf of quantum differential operators on a quantum flag manifold are given by the quantum group. As corollaries we retrieve Joseph and Letzter's quantum versions of classical enveloping algebra theorems of Duflo and Kostant. We also describe the center of the ad-integrable part of the quantum group and the adjoint Lie algebra action on it.