# Quasi-polynomials and the singular $[Q,R]=0$ theorem

## TL;DR

This paper revisits the `shift-desingularization' version of the $[Q,R]=0$ theorem for singular symplectic quotients, using quasi-polynomial behavior and index formulas to analyze multiplicities in geometric quantization.

## Contribution

It introduces an approach combining the Szenes-Vergne proof with Berline-Vergne index formula to analyze multiplicities for singular quotients, extending previous methods.

## Key findings

- Demonstrates quasi-polynomial behavior of multiplicities
- Adapts stationary phase expansion for singular cases
- Provides a new perspective on the $[Q,R]=0$ 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 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.