This article is a sequel to [6]. 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 [12] and Dubrovin [4]. The other is the natural Saito constructed by Kato-Mano-Sekiguchi [5] and by Arsie-Lorenzoni [1]. We study the relationship between these two Saito structures from the viewpoint of almost duality.on the orbit spaces of the duality groups [5]. Arsie and Lorenzoni also studied the same Saito structures for the duality groups of rank n = 2, 3 [1]. In [6], we formulated the almost duality for the Saito structure and characterized their Saito structure. We call it the natural Saito structure because it comes from the trivial connection.So the orbit space of a finite Coxeter group or a Shephard group has both the CS Frobenius structure and the natural Saito structure. A natural question is that whether 2010 Mathematics Subject Classification. Primary 53D45; Secondary 20F55.

We prove that the Frobenius structure constructed from the Gromov-Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the Witten-Dijkgraaf-Verlinde-Verlinde equations with certain natural initial conditions.

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We associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito-Takebayshi and Yamada for elliptic Weyl groups and elliptic Artin groups to the Weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems. Moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of 2-Calabi-Yau completion of the bound quiver algebra.