Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification. 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 $k$ defined on $G^\mathbb{C}$.