Quantization commutes with singular reduction: cotangent bundles of compact Lie groups

TL;DR

This paper proves that in the case of cotangent bundles of compact Lie groups, geometric quantization commutes with symplectic reduction even when the quotient is singular, extending the Guillemin-Sternberg conjecture.

Contribution

It demonstrates that quantization commutes with reduction for singular quotients of cotangent bundles of compact Lie groups, using Dolbeault-Dirac operators in geometric quantization.

Findings

Quantization reproduces Hall's Hilbert space from 2002.

Quantization of the singular quotient matches the reduced space's Hilbert space.

The approach extends the Guillemin-Sternberg conjecture to singular cases.

Abstract

We analyse the `quantization commutes with reduction' problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin-Sternberg Conjecture) for the conjugate action of a compact connected Lie group G on its own cotangent bundle T*G. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden-Weinstein quotient) T*G/Ad G is typically singular. In the spirit of (modern) geometric quantization, our quantization of T*G (with its standard Kaehler structure) is defined as the kernel of the Dolbeault-Dirac operator (or, alternatively, the spin Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of T*G reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarisation. We then define the quantization of the singular quotient T*G/Ad G$ as the kernel of the (twisted) Dolbeault-Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space L^2(T)^{W(G,T)}.