Local Cohomology of Modular Invariant Rings

For K a field, consider a finite subgroup G of \({\text {GL}}_n(K)\) 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.