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.



Symmetries of the flag variety
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 $b\_2=1$).


