Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles 
$$
 \mathcal{P} = G\times\_P \mathfrak{p} \subset G/P \times \mathfrak{gl}
$$
and
$$
 \mathcal{T}\_{G/P} = G\times\_P \mathfrak{g/p}.
$$


I would like to understand the space of sections of the latter, eventually in a way which works if $G$ is any principal bundle.


Bott showed that there is an exact sequence
$$
0\rightarrow H^0(G/P,\mathcal{P}) \rightarrow H^0(G/P,\mathcal{O}\_{G/P}\otimes\mathfrak{g})\rightarrow H^0(G/P,\mathcal{P})\rightarrow H^1(G/P,\mathcal{P})\rightarrow 0
$$


Question 1: If $G/P = \mathbb{P}^{1}$ then $H^0(\mathbb{P}^{1}, \mathcal{T}\_{\mathbb{P}^{1}}) = \operatorname{End}(\mathbb{C}^{2})/\mathbb{C} I$. How does $\mathcal{P}$ split as a direct sum of line bundles on $\mathbb{P}^1$? 


Question 2: Is there a vanishing theorem for $H^1(G/P,\mathcal{P})$ for other flag varieties? 


Related question: [cohomology of tangent bundle](https://mathoverflow.net/questions/122415/cohomology-of-tangent-bundle)



What is the cohomology of the tangent bundle of a flag variety?
The answer might be in


Michel Demazure. Automorphismes et déformations des variétés de Borel. Invent. Math., 39(2):179–186, 1977


