D-modules on representations of Capelli type

We prove that the category of regular holonomic DV-modules invariant under the action of G′ is equivalent to the category of graded modules of finite type over a suitable algebra A, except for few special cases. Indeed the Levasseur's conjecture [28, Conjecture 5.17, p. 508] fails in these cases because of the disconnectedness of the stabilizers of some “smaller” orbits.