Let G be a polyhedral group, namely a finite subgroup of
SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau
resolution Y of the polyhedral singularity C^3/G. The classical McKay
correspondence describes the classical geometry of Y in terms of the
representation theory of G. In this paper we describe the quantum
geometry of Y in terms of R, an ADE root system associated to G.
Namely, we give an explicit formula for the Gromov-Witten partition
function of Y as a product over the positive roots of R. In terms of
counts of BPS states (Gopakumar-Vafa invariants), our result can be
stated as a correspondence: each positive root of R corresponds to one
half of a genus zero BPS state. As an application, we use the CRC to
provide a full prediction for the orbifold Gromov-Witten invariants of
[C^3/G].