How do we give G/T a symplectic structure I am new in those staff and I can't find anything introductory explaining those stuff. I am interested in the following. Given $G$ a compact Lie group and consider it's maximal torus $T$. Then I have read the $G/T$ can be given a symplectic structure. Can anyone explain me how this is achieved or reference me to an introduction on those stuff.
Thanks in advance
 A: One way to see this is to consider the more general fact that coadjoint orbits carry a canonical symplectic structure, and that the 'generic' semisimple coadjoint orbit is diffeomorphic to $G/T$.
If $G$ is a Lie group with Lie algebra ${\mathfrak g}$, it naturally acts on ${\mathfrak g}^{\ast}$ by $(g.\varphi)(X) := \varphi(\text{Ad}(g^{-1})(x))$. The stabilizer of $\varphi\in{\mathfrak g}^{\ast}$ under this action has Lie algebra $\text{stab}_\varphi = \{X\in{\mathfrak g}\ |\ \varphi([X,-])\equiv 0\}$, and the tangent space of the orbit ${\mathscr O}_\varphi\subset {\mathfrak g}^{\ast}$ is naturally isomorphic to ${\mathfrak g}/\text{stab}_\varphi$. This space has a canonical bilinear form given by $\omega(\overline{X},\overline{Y}) := \varphi([X,Y])$, which is well-defined and non-degenerate by the description of $\text{stab}_\varphi$ and anti-symmetric by the anti-symmetry of $[-,-]$ - in other words it's a symplectic form. It can be checked that is canonically extends to a $G$-invariant form on ${\mathscr O}_\varphi$, so ${\mathscr O}_\varphi$ is a symplectic manifold.
Knowing this, it suffices to check that we can find $\varphi$ such that $\text{stab}_\varphi={\mathfrak t}$. Let's look at the complexified situation first: ${\mathfrak g}_{\mathbb C}$ has a root space decomposition with respect to the Cartan subalgebra ${\mathfrak h} := {\mathfrak t}_{\mathbb C}$. Denoting the projection of ${\mathfrak g}_{\mathbb C}$ onto ${\mathfrak h}$ wrt. this decomposition by $\pi$, any $\gamma\in{\mathfrak h}^{\ast}$ gives rise to $\varphi_\gamma := \gamma\circ\pi\in {\mathfrak g}^{\ast}$. Also $$\text{stab}_{\varphi_\gamma}={\mathfrak h}\oplus\bigoplus_{\substack{\alpha\in\Phi\\ \gamma(h_\alpha)=0}} {\mathfrak g}_\alpha,$$ so we get $\text{stab}_{\varphi_\gamma}$ whenever $\gamma$ it in the complement of the finitely hyperplanes $\{\gamma\ |\ \gamma(h_\alpha)=0\}$ - an open, dense set. Finally, one needs to check that ${\mathfrak t}_{\mathbb R}^{\ast}\subset{\mathfrak h}_{\mathbb R}^{\ast}$ is not contained in any of these complex hyperplanes, to deduce that in the real setting we also have $\text{stab}_{\varphi_\gamma}={\mathfrak t}$ for generic $\gamma\in{\mathfrak t}^{\ast}$.
I have left out several details here - if you want to dig into them, just ask, or have a look in the very well-written book Nilpotent Orbits in Semisimple Lie algebras
