Differential calculus on quantized irreducible flag manifolds

We give a new description of the differential calculus on the quantized irreducible flag manifold given by Heckenberger-Kolb. We also define a quantum analogue of the Dolbeault complex associated to an equivariant vector bundle on the irreducible flag manifold, and show that its cohomology groups coincide with the cohomology groups given by Andersen-Polo-Wen defined using the induction functor.