Superspace coinvariants and inverse systems for $GL_n(\mathbb{F}_q)$

Let $q$ be a prime power and write $\Omega$ for the bigraded algebra of regular differential forms over $\mathbb{F}_q^n$. The general linear group $GL_n(\mathbb{F}_q)$ acts on $\Omega$; write $SI \subseteq \Omega$ for the ideal generated by $GL_n(\mathbb{F}_q)$-invariants with vanishing constant term. The {\em $GL_n(\mathbb{F}_q)$-superspace coinvariant ring} is the quotient $SR := \Omega/SI$. We calculate the bigraded Hilbert series of $SR$ and give an operator-theoretic characterization of the inverse system $SI^\perp$. Our results extend to subgroups $G$ of $GL_n(\mathbb{F}_q)$ which contain $SL_n(\mathbb{F}_q)$.