Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its [complexification][1].
I am looking for a symplectic structure (without use of coordinates) on 
$$
Sym^kG^{\mathbb{C}},
$$
i.e. on the space of all symmetric tensors of order $k$ defined on $G^\mathbb{C}$.

  [1]: http://en.wikipedia.org/wiki/Complexification_%28Lie_group%29