  In arXiv:2004.01871 Satake introduced the notions of admissible triplets and
good basic invariants for finite complex reflection groups. For irreducible
finite Coxeter groups, he showed the existence and the uniqueness of good basic
invariants. Moreover he showed that good basic invariants are flat in the sense
of K.Saito's flat structure. He also obtained a formula for the multiplication
of the Frobenius structure. In this article, we generalize his results to
finite complex reflection groups. We first study the existence and the
uniqueness of good basic invariants. Then for duality groups, we show that good
basic invariants are flat in the sense of the natural Saito structure
constructed in arXiv:1612.03643. We also give a formula for the potential
vector fields of the multiplication in terms of the good basic invariants.
Moreover, in the case of irreducible finite Coxeter groups, we derive a formula
for the potential functions of the associated Frobenius manifolds.
