The Koszul complex of a moment map

Let $K\to U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $ρ\colon V\to\mathfrak k^*$. We have the Koszul complex ${\mathcal K}(ρ,\mathcal C^\infty(V))$ of the component functions $ρ_1,...,ρ_k$ of $ρ$. Let $G=K_{\mathbb C}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $ρ^{-1}(0)$ 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 $ρ$ be a moment mapping and consider the Koszul complex given by the component functions of $ρ$. We show that the Koszul complex is a resolution of the smooth functions on $Z=ρ^{-1}(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is 1-large.

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