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 $\mathrm{rad}: \mathcal{D}(V)^G\to A_κ$ from the $G$-invariant differential operators on $V$ to the spherical subalgebra $A_κ$ 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 $A_κ$ is a simple ring. When $A_κ$ is simple, we also show that the kernel of $\mathrm{rad}$ is $\left(\mathcal{D}(V)τ(\mathfrak{g}\right)^G$, where $τ:\mathfrak{g}\to \mathcal{D}(V)$ is the differential of the $G$-action.

Theorems & Definitions (46)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Definition 2.1
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.9
• proof