Semisimple module categories with fusion rules of the compact full flag manifold type

We classify semisimple left module categories over the representation category of a type A quantum group whose fusion rules arise from the maximal torus. The classification is connected to equivariant Poisson structures on compact full flag manifolds in the operator-algebraic setting, and on semisimple coadjoint orbits in the algebraic setting. We also provide an explicit construction based on the BGG categories of deformed quantum enveloping algebras, whose unitarizability corresponds to being of quotient type. Finally, we present a brief discussion of the non-quantum case.