Geometric quantization for proper actions. (English) Zbl 1211.53101
The main purpose of this paper is to generalize the Guillemin-Sternberg geometric quantization conjecture [V. Guillemin and S. Sternberg, Invent. Math. 67, 515–538 (1982; Zbl 0503.58018)] to the case of non-compact spaces and group actions. For this purpose, a \(G\)-invariant index is generalized to the non-compact case and the main result, which might be thought of as a “quantization commutes with reduction” result, solves a conjecture of P. Hochs and N. P. Landsman [J. K-Theory 1, No. 3, 473–533 (2008; Zbl 1159.19004)]. Ulrich Bunke presents in an Appendix the \(KK\)-theoretic interpretation of the index introduced by the authors.
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
| 53D50 | Geometric quantization |
| 53D20 | Momentum maps; symplectic reduction |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
Keywords:
geometric quantization; locally compact groups; Hochs-Landsman conjecture; Guillemin-Sternberg conjecture; equivariant \(K\)-theory; index theorem for generalized orbifoldsReferences:
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