A unitary "quantization commutes with reduction" map for the adjoint action of a compact Lie group

TL;DR

This paper constructs a natural, unitary map linking the geometric quantizations of a compact Lie group's cotangent bundle and its reduced phase space, demonstrating that quantization commutes with reduction in this setting.

Contribution

It introduces a geometrically natural, unitary map for the adjoint action of a compact Lie group, advancing the understanding of quantization and reduction.

Findings

The map is a constant multiple of a unitary map.

Quantization commutes with reduction for the adjoint action.

The construction uses geometric quantization with half-forms.

Abstract

Let $K$ be a simply connected compact Lie group and $T^{\ast}(K)$ its cotangent bundle. We consider the problem of "quantization commutes with reduction" for the adjoint action of $K$ on $T^{\ast}(K).$ We quantize both $T^{\ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{\ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.