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 $G^{\prime }$ 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 $G^{\prime }$-invariant polynomials on $V$ ($\mathbb{C}[V{]}^{G^{\prime }}=\mathbb{C}[f]$) and that $f\notin \mathbb{C}[V{]}^G$. 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 $\mathcal{A}$.

References in corpus (1)

• Invariant differential operators and an infinite dimensional Howe-type correspondence. Part I: Structure of the associated algebras of differential operators