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.

59 pages; minor typos and references updated

Cited by in corpus (1)

• Invariant holonomic systems on symmetric spaces and other polar representations