  For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$
with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let
$\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants
$R^G$. We study how the local cohomology module $H^n_{\mathfrak{n}}(R^G)$
compares with $H^n_{\mathfrak{n}}(R)^G$. Various results on the $a$-invariant
and on the Hilbert series of $H^n_\mathfrak{n}(R^G)$ are obtained as a
consequence.

Commutative Algebra