Equivariant Symplectic Forms

Abstract

Suppose we are given an action G × M → M. In the Cartan model, an element

$$\tilde \omega \in \Omega _G^2(M) = ({\Omega ^2}{(M)^G} \otimes {S^0}({g^*}) \oplus {({\Omega ^0}(M) \otimes {S^1}({g^*}))^G}$$

can be written as

$$\tilde \omega = \omega - \phi $$

where ω ∈ Ω²(M) is a two-form invariant under G and

$$ \phi \in {({\Omega ^0}(M) \otimes {g^*})^G}$$

can be considered as a G equivariant map,

$$\phi :g \to {\Omega ^0}(M) = F(M)$$

from the Lie algebra, g to the space of smooth functions on M. For each ξ ∈ g, ø (ξ) is a smooth function on M, and this function depends linearly on ξ Therefore, for each m ∈ M, the value ø (ξ(m)) depends linearly on, so we can think of ø as defining a map from M to the dual space g* of the Lie algebra of g:

$$\phi :M \to {g^*},\left\langle {\phi (m),\xi } \right\rangle : = \phi (\xi )(m).$$