Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have 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 $\mathfrak{g}/\mathfrak{h}$?
 A: 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 $x$.
A: Actually, the issue is that there is usually more than one 'natural' symplectic structure on $G/T$, where $T$ is a Cartan subgroup.  The space of $G$-invariant closed $2$-forms on $G/T$ has dimension equal to the rank of $G$ (which is the dimension of $T$), and the generic one is nonsingular, so you might have trouble picking out a 'natural' one from among this class when the rank of $G$ is greater than $1$. 
For example, when $G = \mathrm{SO}(4)$, which has rank $2$, the quotient $G/T$ is $S^2\times S^2$ and you can make the symplectic form take arbitrary nonzero values on each $S^2$-factor.
