Satake's good basic invariants for finite complex reflection groups

TL;DR

This paper extends Satake's theory of good basic invariants from irreducible finite Coxeter groups to all finite complex reflection groups, establishing existence, uniqueness, and flatness properties, and deriving formulas for Frobenius structures.

Contribution

It generalizes Satake's results to finite complex reflection groups, proving existence, uniqueness, and flatness of good basic invariants, and providing explicit formulas for Frobenius structures.

Findings

Existence and uniqueness of good basic invariants for finite complex reflection groups.

Good basic invariants are flat in the sense of Saito's structure for duality groups.

Formulas for potential vector fields and Frobenius manifold potentials.

Abstract

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.