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

Abstract

We prove the following theorem. Let $G$ be a finite group generated by unitary reflections in a complex Hermitian space $V={\mathbb{C}}^{\ell }$ and let $G^{\prime }$ 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 $\xi :\mathcal{H}(G^{\prime })\otimes \mathcal{H}(G{)}^{G^{\prime }}\stackrel{\sim }{⟶}\mathcal{H}$ of graded $\mathcal{N}$-modules, where $\mathcal{N}:=N_{\mathrm{GL}(\mathrm{V})}(G)\cap N_{\mathrm{GL}(\mathrm{V})}(G^{\prime })$ and ${\mathcal{H}}^{G^{\prime }}$ is the space of $G^{\prime }$-fixed points of $\mathcal{H}$. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups.

This version includes an application to reductive groups

References in corpus (1)

• Simple extensions of reflection subgroups of primitive complex reflection groups