Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its [complexification](http://en.wikipedia.org/wiki/Complexification_%28Lie_group%29).
I am looking for a symplectic structure (without use of coordinates) on 
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here $G^{\mathbb{C}}=T^\*G$.(this equality is trivial by polar decomposition in the case, when $G$ is compact Lie group )


i.e. on the space of all [symmetric tensors of order $k$ d](http://en.wikipedia.org/wiki/Symmetric_tensor)efined on $G^\mathbb{C}$.



Symplectic structure on $Sym^kG^{\mathbb{C}} $
Such a thing doesn't exist. The symmetric square of the cotangent bundle of a real $n$-dimensional manifold has dimension $n + {n+1 \choose 2}$, which is in particular odd whenever $n \equiv 2 \bmod 4$. So for example we can take $G = \text{SU}(2), G\_{\mathbb{C}} = \text{SL}\_2(\mathbb{C})$, which has real dimension $6$. 


