Quantum flag varieties, equivariant quantum D-modules, and localization of quantum groups

Let Oq(G) be the algebra of quantized functions on an algebraic group G and Oq(B) its quotient algebra corresponding to a Borel subgroup B of G. We define the category of sheaves on the “quantum flag variety of G” to be the Oq(B)-equivariant Oq(G)-modules and prove that this is a proj-category. We construct a category of equivariant quantum D-modules on this quantized flag variety and prove the Beilinson–Bernstein's localization theorem for this category in the case when q is transcendental.