Let $G$ be a compact Lie Group, and $G^\mathbb{C}$ be its [complexification][1], then I am looking for a symplectic structure (without using of coordinates)on 
$$
Sym^kG^{\mathbb{C}}
$$

the space of all symmetric tensors of order k defined on $G^\mathbb{C}$


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