I am completely a beginner in this field. I wonder know whether there is appropriate notion for quantum flag variety of finite dimensional Lie algebra. If so, what is the correspondent notion for "quantum differential operator on this quantum flag variety". If such notions exist. I think there should be some kind of quantum analogue of Beilinson-Bernstein localization for quantum group (or quantized enveloping algebra)?
The second question is related to the questionD-module theory Scott Carnahan mentioned that the category of D-module can be taken as category of quasi coherent sheaves on DeRham stack. So, if there exits "quantum D-module". Then in this case, the category of D-module can be taken as category of quasi coherent sheaves on "quantized" DeRham stack? How do we define the quantized DeRham stack? Or more generally, is there an appropriate notion for "quantized" stack?