I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be.

Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a projective manifold. A way to show this is to consider the complexification of $G$ and identify $G/T$ as $G_{\mathbb C}/B$.

It seems to me that the following are true:

(1) The root space of $G$ corresponds to $H^{1, 1}\big( G/T\big)\cap H^2(G/T, \mathbb R)$: as $G/T$ is simply connected, $H^2(G/T, \mathbb Z)\cong H_2(G/T, \mathbb Z) \cong \pi_2(G/T)$ and $\pi_2(G/T)$ are generated by those "$\mathbb P^1$" corresponds to each $\alpha \in \Delta^+$ (the $su(2)$ subalgebra $S_\alpha$, to be precise).

(2) The positive Weyl Chamber $C$ corresponds to the Kahler cone of $G/T$, for similar reason as in (1).

So my question is, is (1) and (2) actually holds?

For the following two, I am not so sure. It is related to the proof of Borel Weil Bott theorem.

(3) Is $\rho = \frac{1}{2}\sum_{\alpha\in \Delta^+} \alpha$ corresponds to the First Chern class of $G/T$? Because we know that if a integral weight $\lambda$ satisfies $\lambda + \rho \in C$, then all higher cohomology $H^p(G/T, L_\lambda)$ ($p>0$) vanishes. If $C$ is the Kahler cone, then it seems that the statement is a consequence of Kodaira vanishing theorem, if $\rho$ actually corresponds to the first Chern class.

(4) When $\lambda + \rho$ is not in $C$, there is some correspondance between $H^p(G/T, L_\lambda)$ and $H^0(G/T, L_\mu)$ for some $\mu$ such that $\mu + \rho \in C$. Is there an intuitive geometric reason behind this?

To sum up a bit, can anyone suggest a reference related to these "geometrical aspect" of $G/T$? Thanks in advance.