Canonical equivariant extensions using classical Hodge theory

In [4], Lin and Sjamaar show how to use symplectic Hodge theory to obtain canonical equivariant extensions of closed forms in Hamiltonian actions of compact connected Lie groups on closed symplectic manifolds which have the strong Lefschetz property. In this paper, we show how to do the same using classical Hodge theory. This has the advantage of applying far more generally. Our method makes use of Green’s operator, but, as we will show in [3], it is often possible to make explicit calculations. For nonabelian compact connected Lie groups, we use the small model, which is much simpler than the Cartan model and which has been shown to be chain homotopy equivalent to the Cartan model by Alekseev and Meinrenken (see [1]). In the abelian case, the two models are the same. The final section, however, considers the Cartan model.