I have the feeling (regarding the case of $\mathbb{P}^n$) that the cohomology $H^2(G/P,\mathbb{Q})$ can be spanned by $[D_i]$ where $D_i$ are the zero "divisor" of some holomorphic vector fields $X_i$, spanning $TG/P$ everywhere.
I think this is maybe implicitly done in [I.Bernstein,I. Gel'fand, "Schubert Cells and the Cohomology of Spaces G/P"], but i'm not sure.
On the simplest example of $G/P = \mathbb{P}^n$, i take $X_0 \in \mathfrak{m} \backslash 0$ (where $\mathfrak{m}$ the supplement of $\mathfrak{p} = Lie(P)$ in $\mathfrak{g} = Lie(G)$ spanned by the $X_\alpha$, $\alpha \in \Lambda_{P}$ the complementary roots). Then I "choose" $\tau_i \in W(G)$ to be : $$\tau_i \cdot \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ \vdots \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}$$ and sections : $$ \begin{array}{ccccc} \sigma_i &:&U_i &\simeq& \mathfrak{m} \\&& x & \longrightarrow & exp_{\tau_i}^{-1}(x) \end{array}$$ which correspond to the classical affine charts on $\mathbb{P}^n$.
Fix $X_0 \in \mathfrak{m}\backslash 0$.Then we can define $X_i(\cdot) = \sigma_i^* \tilde{X}_0$ which correspond (if i'm not mistaken) to some translation in each affine coordinates, and vanish exactly on $(X_i)_0 =[\underset{j\neq i}{\oplus} \mathbb{C}e_j]$ .
Now it looks like each $(X_i)_0$ is a left translation by some $w_i \in W(G)$ of a fixed Schubert cell.
- Is it true ?
- Is it general for flag manifolds?
Thanks,