The Koszul complex of a moment map

Abstract

Let $K \to \operatorname{U}(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\rho \colon V \to {\mathfrak k}^*$. We have the Koszul complex $\mathcal{K}(\rho, \mathcal{C}^\infty(V))$ of the component functions $\rho_1, \dots, \rho_k$ 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 $G \to \operatorname{GL}(V)$ is $1$-large, a concept introduced in [11,12]. 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.