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
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