Abstract In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic $D$-modules for rings of invariants of finite groups in characteristic zero, and for strongly $F$-regular finitely generated graded algebras with finite $F$-representation type in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic $D$-modules, in this context, have finite length, and we prove the existence of Bernstein–Sato polynomials in characteristic zero. We obtain these results using a more general version of Bernstein filtrations.

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