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 $G'$ be any reflection subgroup of G. Let ${\mathcal {H}}={\mathcal {H}}(G)$ be the space of G-harmonic polynomials on V. There is a degree preserving isomorphism $\mu :{\mathcal {H}}(G')\otimes {\mathcal {H}}(G)^{G'}\overset{\sim }{{\longrightarrow \;}}{\mathcal {H}}(G)$ of graded ${\mathcal {N}}$-modules, where ${\mathcal {N}}:=N_{{\text {GL}}(V)}(G)\cap N_{{\text {GL}}(V)}(G')$ and ${\mathcal {H}}(G)^{G'}$ is the space of $G'$-fixed points of ${\mathcal {H}}(G)$. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups. An application is given to counting rational conjugates of reductive groups over ${\mathbb {F}}_q$.