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

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 a 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.