Bundles of pseudodifferential operators and modular forms (Q1582319)

The author of this interesting paper studies connections between pseudodifferential operators and modular forms. He uses vector bundles over a Riemann surface for which the fibers are spaces of particular pseudodifferential operators. More precisely, the author works with pseudodifferential operators on the upper half-plane \(\mathcal H\) defined as \(\{z\in \mathbb{C}: I m(z) > 0\}\). If \(\mathcal F\) indicates the ring of complex-valued analytic functions defined on \(\mathcal H\) and \(\partial =\partial_z\) denotes the differentiation operator acting on \(\mathcal F\), then a pseudo-differential operator is a formal Laurent series \(\sum^{n_0}_{n=-\infty}\xi_n(z)\partial^n\) where \(\xi_n\in{\mathcal F}\), \(n_0\in \mathbb{Z}\), and \(\partial^{-1}\) denotes the formal inverse of the operator \(\partial\). The author denotes with \(\Psi DO({\mathcal F})\) the class of all such pseudodifferential operators. Using Leibniz's rule, one defines a multiplication among these operators and thus, \(\Psi DO({\mathcal F})\) becomes a ring. The author establishes a right action of the group \(\text{SL}(2,\mathbb{R})\) on the subset \(\Psi DO({\mathcal F})_w\) of \(\Psi DO({\mathcal F})\) that consists of all the formal Laurent series \(\sum^w_{n=-\infty}\xi_n(z)\partial^n\). Next, the author constructs fiber bundles over a quotient of \(\mathcal H\) by a discrete subgroup of \(\text{SL}(2,\mathbb{R})\) in such a way that the fibers are isomorphic to the spaces \(\Psi DO({\mathcal F})_w\). To establish the connection between modular forms and the classes \(\Psi DO({\mathcal F})_w\) the author introduces the operator \({\mathcal L}_m:{\mathcal F}\to\Psi DO({\mathcal F})_{-m}\) defined as \[ {\mathcal L}_m(f)={2m-1\choose m}\sum^\infty_{n=0}(-1)^n\;\frac{(n + m)!(n+m-1)!}{n!(n+2m-1)!} f(z)^{(n)}\partial^{-m-n}. \] This operator has the property \[ {\mathcal L}_m(f|_{2m}[\gamma]) ={\mathcal L}_m(f)\cdot\gamma \] for each positive integer \(m\) and for all \(f \in{\mathcal F}\) and \(\gamma\in \text{SL}(2,\mathbb{R})\). By \(f|_{2m}[\gamma]\) the author denotes the complex function defined on \(\mathcal H\) as \((cz + d)^{-k}f(\gamma z)\) where \(\gamma=\left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix}\right) \in \text{SL} (2,\mathbb{R})\).