Nonlinear symplectic Grassmannians and Hamiltonian actions in prequantum line bundles (Q2893466)

Let \((M,\omega)\) be a finite dimensional symplectic manifold with a Hamiltonian action of a Lie group \(G\), with equivariant moment map \(\mu: M\to {\mathcal G}^*\). Given \(x\in M\), it is known that the isotropy group \(G_x\) acts linearly on \((T_x M,\omega)\) in a Hamiltonian fashion, with moment map equal to the Taylor expansion of \(\mu\) at \(0\in T_x M\) of order \(2\).NEWLINENEWLINEThe author proves an analogue in a more general, infinite-dimensional setting:NEWLINENEWLINENEWLINE Theorem 1. If \((M,\Omega)\) is now a Fréchet manifold with weakly regular symplectic structure, and \(G\) is a regular Fréchet-Lie group, the same fact holds.NEWLINENEWLINEThe author uses this result to give an explanation of a formula by \textit{S. K. Donaldson} [J. Differ. Geom. 59, No. 3, 479--522 (2001; Zbl 1052.32017)], which describes the momentum map of the Hamiltonian infinitesimal action of the Lie algebra of the group of Hamiltonian diffeomorphisms of a closed integral symplectic manifold, on sections of its prequantum line bundle.