Geometric Quantization on a Coset Space G/H

TL;DR

This paper explores geometric quantization on coset spaces, showing how different quantizations relate to symplectic forms and irreducible representations, connecting to Wong's equation and Weil's theorem.

Contribution

It demonstrates how inequivalent geometric quantizations on G/H can be derived using symplectic forms and irreducible representations, linking to established theorems.

Findings

Inequivalent quantizations correspond to specific symplectic 2-forms.

Irreducible representations of H label different quantizations.

Weil's theorem guarantees the existence of a Hermitian bundle over G/H.

Abstract

Geometric quantization on a coset space $G/H$ is considered, intending to recover Mackey's inequivalent quantizations. It is found that the inequivalent quantizations can be obtained by adopting the symplectic 2-form which leads to Wong's equation. The irreducible representations of $H$ which label the inequivalent quantizations arise from Weil's theorem, which ensures a Hermitian bundle over $G/H$ to exist.