Koszul properties of the moment map of some classical representations

This work concerns the moment map $\mu$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $S\mathrm{/}(\mu )$, the coordinate algebra of the zero fibre of $\mu$, be Koszul. The main result is that this algebra is not Koszul for the standard representation of ${\mathfrak{s}\mathfrak{l}}_n$, and of ${\mathfrak{s}\mathfrak{p}}_n$. This is deduced from a computation of the Betti numbers of $S\mathrm{/}(\mu )$ as an $S$-module, which are of interest also from the point of view of commutative algebra.