Let $X = G/P$ a rational homogeneous variety, e.g. a grassmannian. Consider a short exact sequence $$ 0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow 0$$ where $E_i$ are homogeneous vector bundles and the maps preserve the action of $G$.
Consider the associated long exact sequence in cohomology $$ 0 \longrightarrow H^0(E_1) \longrightarrow H^0(E_2) \longrightarrow H^0(E_3) \longrightarrow H^1(E_1) \longrightarrow \cdots$$ By Bott's Theorem all the cohomology groups are representations of $G$.
My question is whether the maps between cohomology groups are morphisms of representations.
I believe that, if true, a way to prove it is showing that the derived functors of $$ H^0 : \mathrm{HomogeneousVectorBundles} \longrightarrow \mathrm{Vect} $$ and $$ H^0 : \mathrm{HomogeneousVectorBundles} \longrightarrow \mathrm{Rep}(G) $$ are the same modulo the obvious forgetuful functor $\mathrm{Rep}(G) \longrightarrow \mathrm{Vect}$. (Here $\mathrm{Vect}$ is the category of finite dimensional vector spaces, $\mathrm{Rep}(G)$ is the category of finite dimensional representations of $G$).
