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.