# Semi-classical analysis of piecewise quasi-polynomial functions and applications to geometric quantization

## TL;DR

This paper develops a semi-classical analysis framework for piecewise quasi-polynomial functions, revealing their asymptotic behavior and applications to geometric quantization, including a new proof of functoriality in formal quantization.

## Contribution

It introduces a novel asymptotic expansion for distributions associated with piecewise quasi-polynomial functions, connecting to geometric quantization and functorial properties.

## Key findings

- Asymptotic expansion generalizes Euler-Maclaurin formula
- Unique determination of functions by asymptotic expansions
- Simplified proof of functoriality in formal quantization

## Abstract

Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $\Theta(m;k)$ associated to a piecewise quasi-polynomial function $m$. The family is indexed by an integer $k \in \mathbb{Z}_{>0}$, and admits an asymptotic expansion as $k \rightarrow \infty$, which generalizes the expansion obtained in the Euler-Maclaurin formula. When $m$ is the multiplicity function arising from the quantization of a symplectic manifold, the leading term of the asymptotic expansion is the Duistermaat-Heckman measure. Our main result is that $m$ is uniquely determined by a collection of such asymptotic expansions. We also show that the construction is compatible with pushforwards. As an application, we describe a simpler proof that formal quantization is functorial with respect to restrictions to a subgroup.