Invariant theory which uses equivariant sheaves

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Let G be a reductive group over a field of positive characteristic, acting on a polynomial ring S linearly. We proved that if S has a good filtration as a representation of G, then for any parabolic subgroup P of G, the invariant subring S^{U_P} under the action of the unipotent radical U_P of P is a finitely generated strongly F-regular UFD, in particular, it is Gorenstein. We also discussed properties of commutative rings of positive characteristic, and jointly with Mitsuhiro Miyazaki, we discussed G-prime and G-primary G-ideals. These are studies in commutative algebra and invariant theory.