Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 090, 15 pages      arXiv:1907.06113      https://doi.org/10.3842/SIGMA.2019.090

Quasi-Polynomials and the Singular $[Q,R]=0$ Theorem

Yiannis Loizides
Pennsylvania State University, USA

Received July 16, 2019, in final form November 13, 2019; Published online November 18, 2019

Abstract
In this short note we revisit the 'shift-desingularization' version of the $[Q,R]=0$ theorem for possibly singular symplectic quotients. We take as starting point an elegant proof due to Szenes-Vergne of the quasi-polynomial behavior of the multiplicity as a function of the tensor power of the prequantum line bundle. We use the Berline-Vergne index formula and the stationary phase expansion to compute the quasi-polynomial, adapting an early approach of Meinrenken.

Key words: symplectic geometry; Hamiltonian $G$-spaces; symplectic reduction; geometric quantization; quasi-polynomials; stationary phase.

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