The Koszul complex of a moment map

Abstract

Let $K\to U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\rho :V\to {\mathfrak{k}}^{\ast }$. We have the Koszul complex $\mathcal{K}(\rho ,{\mathcal{C}}^{\infty }(V))$ of the component functions ${\rho }_1,...,{\rho }_k$ of $\rho$. Let $G=K_{\mathbb{C}}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on ${\rho }^{-1}(0)$ if and only if $G\to\GL(V)$ is 1-large, a concept introduced in earlier work of the second author. Now let $M$ be a symplectic manifold with a Hamiltonian action of $K$. Let $\rho$ be a moment mapping and consider the Koszul complex given by the component functions of $\rho$. We show that the Koszul complex is a resolution of the smooth functions on $Z={\rho }^{-1}(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is 1-large.

8 pages, final version, to appear in Journal of Symplectic Geometry