分段拟多项式函数的半经典分析及其在几何量子化中的应用
Semi-classical analysis of piecewise quasi-polynomial functions and applications to geometric quantization
摘要：
受指数理论中重数公式应用的启发，我们研究了一类与分段拟多项式函数$m$相关的分布族$Θ(m;k)$.该族由整数$k \in \mathbb{Z}_{>0}$索引，并且允许渐近展开为$k \rightarrow \infty$，其推广了Euler-Maclaurin公式中得到的展开。当$m$是由辛流形的量子化产生的重数函数时，渐近展开的首项是Duistermaat-Heckman测度。我们的主要结果是，$m$是唯一确定的收集这样的渐近展开。我们还表明，建设是兼容的向前推进。作为一个应用，我们描述了一个简单的证明，正式量化是函子方面的限制子群。
英文摘要：
Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $Θ(m;k)$ 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.
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