Let $G$ be a semi-simple algebraic group over $\mathbb C$, $B$ be a Borel and P is be parabolic containing $B$ and let $\mathcal L$ be an ample line bundle on $G/P$. Is it true that $H^0(G/B, L^{\otimes d})=H^0(G/P, L^{\otimes d})$ for all d > 0?


From the projection map from $G/B \rightarrow G/P$ we get an injection from $H^0(G/P, L^{\otimes d})$ to $H^0(G/B, L^{\otimes d})$. But why is it a surjection ?



Global sections of G/B and G/P
Q. Why is it a surjection? 
A. It's a nonzero $G$-equivariant map to (by Borel-Weil) a $G$-irrep.


(Of course this has punted the difficulty, solved geometrically by the commenters, into the Borel-Weil theorem.)


