Quantum Hamiltonian Reduction for Polar Representations

Abstract

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_{\kappa}$ from the $G$-invariant differential operators on $V$ to the spherical subalgebra $A_{\kappa}$ of a rational Cherednik algebra. Under mild hypotheses $\mathrm{rad}$ is shown to be surjective. If $V$ is a symmetric space, then $\mathrm{rad}$ is always surjective, and we determine exactly when $A_{\kappa}$ is a simple ring. When $A_{\kappa}$ is simple, we also show that the kernel of $\mathrm{rad}$ is $\left(\mathcal{D}(V)\tau(\mathfrak{g}\right)^G$, where $\tau:\mathfrak{g}\to \mathcal{D}(V)$ is the differential of the $G$-action.