On Embedding the 1 : 1 : 2 Resonance Space in Apoisson Manifold

The Hamiltonian actions of S 1 on the symplectic manifold R 6 in the 1 : 1 : ?2 and 1 : 1 : 2 resonances are studied. Associated to each action is a Hilbert basis of polynomials deening an embedding of the orbit space into a Euclidean space V and of the reduced orbit space J ?1 (0)=S 1 into a hyperplane V J of V , where J is the quadratic momentum map for the action. The orbit space and the reduced orbit space are singular Poisson spaces with smooth structures determined by the invariant functions. It is shown that the Poisson structure on the orbit space, for both the 1 : 1 : 2 and the 1 : 1 : ?2 resonance, cannot be extended to V , and that the Poisson structure on the reduced orbit space J ?1 (0)=S 1 for the 1 : 1 : ?2 resonance cannot be extended to the hyperplane V J. 1. Introduction In this paper we study certain singular Poisson spaces arising from Hamiltonian actions of Lie groups on symplectic manifolds. The singular Poisson spaces are embedded, using Hilbert maps, into Euclidean spaces and we prove that the singular Poisson structure cannot be extended to the Euclidean spaces. This disproves a conjecture raised by Cushman and Weinstein 12, 13]. For a Hamiltonian action of a Lie group G on a symplectic manifold (M; !) with an equivariant momentum map J, Marsden and Weinstein deene in 5] the reduced orbit space M , for a value in the dual of the Lie algebra of G. The space M is the quotient space J ?1 ()=G where G is the isotropy group of with respect to the coadjoint action of G. For weakly regular values of J, if G acts freely and properly on the manifold J ?1 (), M is a manifold and there is a unique symplectic