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 $${\mathbb {F}}_q$$.