Chari and Pressley’s A guide to quantum groups, or the original work by Vaksman and Soibelman in 1989, explains that, similarly to the orbit method which relates quantised coadjoint orbits to unitary irreducible representations, there is a relation between dressing orbits of a compact Lie group $G$ and the representation theory of the corresponding quantum group $U_q(g)$.
Does this remain true when $q$ is a root of unity?
