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

TL;DR

This paper explores the relationship between two types of Saito structures on orbit spaces of Coxeter and Shephard groups, focusing on their almost duality, to deepen understanding of their geometric and algebraic properties.

Contribution

It investigates the connection between Frobenius-based and naturally constructed Saito structures on these groups' orbit spaces, highlighting their almost duality relationship.

Findings

Identification of the duality relationship between the two Saito structures

Insights into the geometric structures of Coxeter and Shephard groups

Enhanced understanding of Frobenius structures in relation to Saito structures

Abstract

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.