D-modules on G-representations

Abstract: We give an answer to the abstract Capelli problem: Let $(G, V)$ be a multiplicity-free finite-dimensional representation of a connected reductive complex Lie group $G$ and $G'$ be its derived subgroup. Assume that the categorical quotient $V//G$ is one dimensional, i.e., there exists a polynomial $f$ generating the algebra of $G'$-invariant polynomials on $V$ ($\mathbb{C}[V]^{G'} = \mathbb{C}[f]$) and that $f \not\in \mathbb{C}[V]^{G}$. We prove that the category of regular holonomic $\mathcal{D}_{V}$-modules invariant under the action of $G$ is equivalent to the category of graded modules of finite type over a suitable algebra $\mathcal{A}$.