Generic simplicity of quantum Hamiltonian reductions

TL;DR

This paper proves that under certain generic conditions, the quantum Hamiltonian reduction of differential operators on a smooth affine variety with a reductive group action is simple, highlighting a broad class of cases with this property.

Contribution

It establishes the simplicity of quantum Hamiltonian reductions for very generic characters under flatness and freeness conditions, generalizing previous results in the field.

Findings

Quantum Hamiltonian reductions are simple for very generic characters.

Flatness of the moment map is a key condition.

Freeness of the group action on the inverse image of a generic character.

Abstract

Let a reductive group $G$ act on a smooth affine complex algebraic variety $X.$ Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mu:T^*(X)\to \mathfrak{g}$ be the moment map. If the moment map is flat, and for a generic character $\chi:\mathfrak{g}\to\mathbb{C}$, the action of $G$ on $\mu^{-1}(\chi)$ is free, then we show that for very generic characters $\chi$ the corresponding quantum Hamiltonian reduction of the ring of differential operators $D(X)$ is simple.