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 $$ Sym^kG^{\mathbb{C}}, $$ i.e. on the space of all symmetric tensors of order $k$ defined on $G^\mathbb{C}$.
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Symplectic structure on $Sym^kG^{\mathbb{C}} $
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