Semi-classical analysis of piecewise quasi-polynomial functions and applications to geometric quantization (Q2219491)

In this memorial paper dedicated to Hans Duistermaat the authors study the asymptotic expansion of distributions arising from approximating test functions over polyhedra by the Riemann sum of their values in lattice points. Functorial properties of these expansions as well as their relationship with geometric quantization are discussed, too. More precisely, let \(V\) be a finite-dimensional vector space, \(\Lambda\subset V\) be a lattice and \(P\subset V\) be a rational polyhedron. Let \(\varphi :V\rightarrow{\mathbb R}\) be a smooth test function and assume that we want to study \(\varphi\) along \(P\). An approximation is provided by the Riemann sum \(\sum_{\lambda\in P\cap\Lambda}\varphi (\lambda )\) and this is certainly better if \(\Lambda\) is denser. The Euler-Maclaurin formula gives rise to an asymptotic expansion of \[ \sum\limits_{\lambda\in kP\cap\Lambda}\varphi\left(\frac{\lambda}{k}\right), \] i.e., the sum of values of \(\varphi\) over \(P\cap k^{-1}\Lambda\) when \(k\rightarrow +\infty\) with leading term \(k^{\dim P}\int_P\varphi\mathrm{d}m_\Lambda\) where \(m_\Lambda\) is the translation-invariant measure on \(V\) normalized with respect to \(\Lambda\). This sum can be written as a pairing \[ \langle \Theta (P,\Lambda ;k),\varphi\rangle \] where \(\Theta (P,\Lambda ;k):=\sum_{\lambda\in P\cap k^{-1}\Lambda} \delta_{\lambda /k}\) is a distribution built up from the sum of Dirac-deltas concentrated on the scaled lattice points. Roughly speaking, in this article, the authors consider similar but weighted distributions of the form \[ \sum\limits_{\lambda\in k^{-1}\Lambda}m(k,\lambda ) \delta_{\frac{\lambda}{k}} \] and seek function classes \(S(\Lambda )\) such that: with weights \(m(k, \lambda )\in S(\Lambda )\) the corresponding weighted distributions admit similar asymptotic expansions (see Theorem 1.1 in the article) whose leading terms carry some information about the weights (see Theorem 1.2 in the article) and finally the expansion itself is functorial under pushforward (see Corollary 6.5 in the article). The function class \(S(\Lambda )\) contains generalized quasi-polynomials (see Definition 3.4 in the article for a precise description). An important example for a weighted distribution of this type is the one associated with the multiplicity function \(m\) arising from the geometric quantization of a symplectic manifold; the leading term of its asymptotic expansion is the Duistermaat-Heckman measure.