Almost duality for Saito structure and complex reflection groups II: the case of Coxeter and Shephard groups

It is known that the orbit spaces of the finite Coxeter groups and the Shephard groups admit two types of Saito structures without metric. One is the underlying structures of the Frobenius structures constructed by Saito and Dubrovin. The other is the natural Saito constructed by Kato-Mano-Sekiguchi and by Arsie-Lorenzoni. We study the relationship between these two Saito structures from the viewpoint of almost duality.