A factorisation theorem for the coinvariant algebra of a unitary reflection group

We prove the following theorem. Let $G$ be a finite group generated by unitary reflections in a complex Hermitian space $V={\mathbb{C}}^{\mathrm{\ell }}$ and let $G^{\mathrm{\prime }}$ be any reflection subgroup of $G$. Let $\mathcal{H}(G)$ be the space of $G$-harmonic polynomials on $V$. There is a degree preserving isomorphism $\xi :\mathcal{H}(G^{\mathrm{\prime }})\otimes \mathcal{H}(G{)}^{G^{\mathrm{\prime }}}\mathcal{H}$ of graded $\mathcal{N}$-modules, where $$\mathcal{N}$:=N_{\rm{GL}(V)}(G)\cap N_{\rm{GL}(V)}(G’)$and$$\mathcal{H}$^{G’}$isthespaceof$G’$-fixedpointsof$$\mathcal{H}$$. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups.