Local cohomology of modular invariant rings

For $K$ a field, consider a finite subgroup $G$ of ${\mathrm{GL}}_n(K)$ 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 $H_{\mathfrak{n}}^n(R^G)$ compares with $H_{\mathfrak{n}}^n(R{)}^G$. Various results on the $a$-invariant and on the Hilbert series of $H_{\mathfrak{n}}^n(R^G)$ are obtained as a consequence.