A note on D-modules on the projective spaces of a class of G-representations

Abstract: Consider $(G, V)$ a finite-dimensional representation of a connected reductive complex Lie group $G$ and $\mathbb{P}\left( V\right) $ 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 $\Gamma\left(V, \mathcal{D}V\right)^G$ 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 $\Gamma\left(V, \mathcal{D}_V\right)^G$ is equal to the image of the center of the universal enveloping algebra of $\mathrm{Lie}(G)$ under the differential $\tau: \mathrm{Lie}(G) \longrightarrow \Gamma\left(V, \mathcal{D}_V\right)$ of the $G$-action. Let $\mathcal{A}$ 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 $\mathcal{D}{\mathbb{P}\left( V\right)}$-modules on the complex projective space $\mathbb{P}\left( V\right) $ and the quotient category of finitely generated graded $\mathcal{A}$-modules modulo those supported by $\left{ 0\right} $. 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 $\mathcal{D}_{\mathbb{P}\left( V\right) }$-modules on the projective space of skew-symmetric matrices.