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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

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    $\begingroup$ I don't understand your first question. As for the second --- in characteristic zero you can use Borel--Bott--Weil to answer it. $\endgroup$ – Sasha Mar 13 '15 at 19:06
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    $\begingroup$ Concerning the first question, the answer is $O \oplus O(-1) \oplus O(-1)$. $\endgroup$ – Sasha Mar 13 '15 at 19:56
  • $\begingroup$ @sasha : since the global sections are 3-dim, there must be a misprint in your comment. $\endgroup$ – BS. Mar 16 '15 at 7:51
  • $\begingroup$ BS. He is talking about the sheaf $\mathfrak{p}$, which you would expect to be rank 3, degree -2 and have vanishing cohomology in this case. $\endgroup$ – Anton Isopoussu Mar 16 '15 at 9:58
  • $\begingroup$ oops! I misread the question. Thanks. $\endgroup$ – BS. Mar 16 '15 at 10:20
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The answer might be in

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

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  • $\begingroup$ Fantastic, thank you Giulio! I don't think I would've found that paper on my own. There is a very nice exposition of the results of Demazure and Bott in the book Dmitri N. Akhiezer, Aspects of Mathematics, vol. E27, Friedr. Vieweg, Braunschweig and Wiesbaden, 1995. I picked up this reference from mathoverflow.net/questions/160292/… $\endgroup$ – Anton Isopoussu Mar 16 '15 at 10:29

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