Inductive Freeness of Ziegler's Canonical Multiderivations for Reflection Arrangements

Let $A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $A^{\mathrm{\prime }\mathrm{\prime }}$ of $A$ to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let $A=A(W)$ be the reflection arrangement of a complex reflection group $W$. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler’s canonical multiplicity on the restriction $A^{\mathrm{\prime }\mathrm{\prime }}$ of $A$ to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if $A^{\mathrm{\prime }\mathrm{\prime }}$ itself is inductively free.