Consider (G, V) a finite-dimensional representation of a connected reductive complex Lie group G and P (V) the projective space of V. Denote by G' the derived subgroup of G and assume that the categorical quotient is one dimensional. In the case where the representation (G, V) is also multiplicity-free, it is known from Howe-Umeda 4 that the algebra of G-invariant differential operators Γ (V, DV) G is a commutative polynomial ring. Suppose that the representation (G, V) satisfies the abstract Capelli condition: (G, V) is an irreducible multiplicity-free representation such that the Weyl algebra Γ (V, DV) G is equal to the image of the center of the universal enveloping algebra of Lie (G) under the differential τ: Lie (G) Γ (V, DV) of the G-action. Let A be the quotient algebra of all G'-invariant differential operators by those vanishing on G'-invariant polynomials. The main aim of this paper is to prove that there is an equivalence of categories between the category of regular holonomic D (ₕ) -modules on the complex projective space P (V) and the quotient category of finitely generated graded A-modules modulo those supported by \ 0\. This result is a generalization of 12, Theorem 3. 4 and of 13, Theorem 8. As an application we give an algebraic/combinatorial classification of regular holonomic D (ₕ) -modules on the projective space of skew-symmetric matrices.
Philibert Nang (Wed,) studied this question.