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 $\mathfrak{g}/\mathfrak{h}$?



Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural symplectic structure?
To add to Robert Bryant's answer, the coadjoint orbits of a Lie group always have a natural symplectic structure, the (Lie-)Kirillov-Kostant(-Souriau) form. These are the symplectic leaves of the natrual Lie-Poisson structure on $\mathfrak{g}^\ast$. For $x \in \mathfrak{g}^\ast$, we can identify the coadjoint orbit $\mathcal{O}\_x$ with $G/H\_x$, where $H\_x$ is the stabilizer of $x$ under the coadjoint action. For $x$ such that $H\_x$ is Cartan (which should happen generically), this endows $G/H$ with a symplectic structure. However, this depends on the choice of $x$. For example, even in the absolute simplest case of $G = SU(2)$, the coadjoint orbits are spheres of different radii, so the orbits $\mathcal{O}\_x$ and $\mathcal{O}\_y$ are not symplectomorphic unless $y$ is conjugate to $x$.


