A reduction theorem for good basic invariants of finite complex reflection groups

TL;DR

This paper introduces a reduction process for good basic invariants in finite complex reflection groups, showing how invariants and duality structures induce similar structures in reflection subquotients under certain conditions.

Contribution

It provides a new reduction theorem for good basic invariants of complex reflection groups, extending understanding of their structure and duality properties.

Findings

Reduction process for good basic invariants described

Induction of invariants in reflection subquotients proven

Examples illustrating the reduction process provided

Abstract

This is a sequel to our previous article arXiv:2307.07897. We describe a certain reduction process of Satake's good basic invariants. We show that if the largest degree $d_1$ of a finite complex reflection group $G$ is regular and if $\delta$ is a divisor of $d_1$, a set of good basic invariants of $G$ induces that of the reflection subquotient $G_{\delta}$. We also show that the potential vector field of a duality group $G$, which gives the multiplication constants of the natural Saito structure on the orbit space, induces that of $G_{\delta}$. Several examples of this reduction process are also presented.