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Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety. Is it true that the obvious map $$ \mathfrak g\to \Gamma (T\mathcal B) $$ from $\mathfrak g$ to the Lie algebra of globally defined algebraic vector fields on $\mathcal B$ is an isomorphism?

Remark:
There are examples where the corresponding statement for $G/P$ is false.

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  • $\begingroup$ I don't understand: isn't $\Gamma$ a module over $C^\infty(G/P)$? How could it be finite dimensional? $\endgroup$
    – Uri Bader
    Apr 7, 2016 at 18:13
  • $\begingroup$ I meant $\Gamma$ to mean algebraic sections (equivalently holomorphic sections). I'll edit the question to clarify. $\endgroup$
    – André Henriques
    Apr 7, 2016 at 18:37

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Yes, this is true. In fact, the homogeneous spaces $G/P$ such that $\mathfrak{g}\rightarrow \Gamma (T_{G/P})$ is not an isomorphism have been classified (see e.g. M. Demazure, Inventiones math. 39, 179-186 (1977)): they are the odd-dimensional projective spaces, the Grassmannian of linear subspaces of maximal dimension in a smooth odd-dimensional quadric, and the 5-dimensional quadric. None of them is of the form $G/B$ (in fact they all have $b_2=1$).

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  • $\begingroup$ To save people some typing: eudml.org/doc/142464 Also, would you happen to know some English reference? $\endgroup$
    – Vít Tuček
    Apr 7, 2016 at 22:13
  • $\begingroup$ I would like to point out that even though Demazure's classification is correct as stated, its proof contains two mistakes. They have been pointed out and corrected in a little note (in german) available on my home page. The first mistake is minor and affects the group of type $G_2$ over fields of arbitrary characteristic (so including the complex numbers). The other mistake requires a major revision for non-simply laced groups in very bad characteristic (2 or 3, respectively). $\endgroup$
    – Friedrich Knop
    Apr 22, 2016 at 14:09
  • $\begingroup$ @Friedrich Knop: Thanks for your comment, this is quite useful to know. $\endgroup$
    – abx
    Apr 23, 2016 at 7:20

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