Proper maps, bordism, and geometric quantization

TL;DR

This paper establishes a new geometric framework linking equivariant maps and vector bundles on manifolds to the character ring of a Lie group, and proves the 'Quantization Commutes with Reduction' conjecture in non-compact cases.

Contribution

It introduces an equivalence relation on triples involving manifolds, bundles, and maps, showing their classes form a group isomorphic to a completion of the character ring, and provides a geometric proof of the conjecture.

Findings

The set of equivalence classes forms an abelian group isomorphic to a completion of R(G).

Provides a geometric proof of the 'Quantization Commutes with Reduction' conjecture in non-compact settings.

Establishes a new link between geometric data and representation theory.

Abstract

Let $G$ be a compact connected Lie group acting on a stable complex manifold $M$ with equivariant vector bundle $E$. Besides, suppose $\phi$ is an equivariant map from $M$ to the Lie algebra $\mathfrak{g}$. We can define some equivalence relation on the triples $(M, E, \phi)$ such that the set of equivalence classes form an abelian group. In this paper, we will show that this group is isomorphic to a completion of character ring $R(G)$. In this framework, we provide a geometric proof to the "Quantization Commutes with Reduction" conjecture in the non-compact setting.