Irreducible quantum group modules with finite dimensional weight spaces. II

We classify the simple quantum group modules with finite dimensional weight spaces when the quantum parameter $q$ is transcendental and the Lie algebra is not of type $G_2$. This is part $2$ of the story. The first part being Irreducible quantum group modules with finite dimensional weight spaces. I (arXiv:1504.07042). In that paper the classification is reduced to the classification of torsion free simple modules. In this paper we follow the procedures used by O. Mathieu to reduce the classification to the classification of infinite dimensional admissible simple highest weight modules. We then classify the infinite dimensional admissible simple highest weight modules and show among other things that they only exist for types $A$ and $C$. Finally we complete the classification of simple torsion free modules for types $A$ and $C$ completing the classification of the simple torsion free modules.