The quantum McKay correspondence for singularities of type D

We prove the quantum McKay correspondence formulae conjectured by J. Bryan and A. Gholampour for the type D (binary) polyhedral groups in SU(2) and SO(3). We use the method of induction by the WDVV equation and from the normal subgroups by J. Bryan and A. Gholampour, and the polynomiality technique developed in this article. We are also based on the validity of the corresponding conjecture for type A groups, which is proved by T. Coates, A. Corti, H. Iritani, and H.-H. Tseng.