Local Cohomology of Modular Invariant Rings

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