Poincaré duality algebras and rings of coinvariants

Let $\varrho :G\hookrightarrow GL(n, \mathbb{F})$ be a faithful representation of a finite group $G$ over the field $\mathbb{F}$. Via $\varrho$ the group $G$ acts on $V=\mathbb{F} ^n$ and hence on the algebra ${\mathbb{F}}[V]$ of homogenous polynomial functions on the vector space $V$. R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field $\mathbb{F}$ has characteristic 0, then ${\mathbb{F}}[V] _G$ is a Poincaré duality algebra if and only if $G$ is a pseudoreflection group. The purpose of this note is to extend this result to the case $\vert G\vert\in \mathbb{F} ^{\times}$ (i.e. the order $\vert G\vert$ of $G$ is relatively prime to the characteristic of $\mathbb{F}$ ).