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Ramsey
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Is Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) hashave a natural symplectic structure?

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Zhaoting Wei
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Is $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) has a natural symplectic structure?

Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$).

My questions is: it is always true that we have a natural symplectic structure on the quotient space $G/H$?

If it is not true, could we consider this weaker version: Let $\mathfrak{g}$ and $\mathfrak{h}$ be complexified Lie algebras of $G$ and $H$ respectively. Is there a natural symplectic structure on the quotient space $\mathfrak{g}/\mathfrak{h}$?