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

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 $\hat{I}´$ is a divisor of $d_1$, a set of good basic invariants of $G$ induces that of the reflection subquotient $G_{\hat{I}}´$. 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 $G_{\hat{I}}´$. Several examples of this reduction process are also presented.

(v2) 31 pages, version to appear in Journal of Algebra; revised following the referee's comments, especially an article by Slodowy is added to the references