Action on sheaf cohomology in Bott-Borel-Weil theorem Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and $G \times _P V$ associated homogeneous vector bundle over $G/P$. Let $\mathcal O _P(\lambda)$ be sheaf holomorphic sections of that bundle.
The Bott-Borel-Weil theorem describes structure of sheaf cohomology $H^r(G/P,\mathcal O _P(\lambda))$ as a $G$-module. (My reference is Baston, Eastwood: The Penrose transform, its interaction with representation theory - chapter 5.)
What I am missing is: How is $G$-action on $H^r(G/P,\mathcal O _P(\lambda))$ apriori defined?
I know that $G$ acts on global sections of $\mathcal O _P(\lambda))$. Is it posible to make some kind of acyclic resolution of $\mathcal O _P(\lambda)$ with $G$-module structure on global sections?
 A: I do not have a complete answer, but it is nevertheless too long for a comment.
If $X$ is a $G$-variety equipped with a $G$-equivariant vector bundle $\xi: E\to X$ (the $G$-equivariant structure in your case is described in Jyrki's comment), then the sheaf of sections $\mathscr F$ of $E$ is a $G$-equivariant quasi-coherent (even locally free) sheaf on $X$ in the sense that, denoting $\rho,\pi: G\times X\to X$ the action resp. projection map, it comes equipped with an isomorphism $\rho^{\ast}{\mathscr F}\cong\pi^{\ast}{\mathscr F}$ that is associative and unital. Denote the category of such sheaves by $\text{QCoh}^G(X)$. In case $X=\text{pt}$, this gives the rational $G$-representations, and any $G$-equivariant morphism $f: X\to Y$ induces a pushforward $\text{QCoh}^G(X)\to\text{QCoh}^G(Y)$ which in case of $Y=\text{pt}$ is the global section functor.
Now two questions have to be addressed: 


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*Does $\text{QCoh}^G(X)$ have enough injectives?


If yes, one can define the right derived functors  $\textbf{R}^{i}f_{\ast}:\text{QCoh}^G(X)\to\text{QCoh}^G(Y)$.


*

*Does $\text{QCoh}^G(X)\to\text{QCoh}(X)$ preserve injectives, or at least map injectives to $f_{\ast}$-acyclics?


If yes, then the equivariant and non-equivariant versions of $\textbf{R}^if_{\ast}$ are compatible. 
In particular, taking $Y=\text{pt}$, the sheaf cohomology of any $G$-equivariant quasi-coherent sheaf admits a canonical structure of a rational $G$-representation, as desired.
Concerning the first question, I found something in Bezrukavnikov, Perverse Coherent Sheaves http://arxiv.org/pdf/math/0005152.pdf: 
The forgetful functor $\varepsilon: \text{QCoh}^G(X)\to\text{QCoh}(X)$ admits an exact right adjoint $$\text{av} := \rho_{\ast}\pi^{\ast}:\text{QCoh}(X)\to\text{QCoh}^G(X)$$ which preserves injectives by exactness of $\varepsilon$. Moreover, $\text{av}$ is exact and the unit morphism $\text{id}\to\text{av}\circ\varepsilon$ is a monomorphism, so given any ${\mathscr F}\in\text{QCoh}^G(X)$, an embedding into an injective, $G$-equivariant quasi-coherent sheaf can be constructed as $${\mathscr F}\hookrightarrow\text{av}(\varepsilon {\mathscr F})\hookrightarrow\text{av}({\mathscr I})$$ where $\varepsilon{\mathscr F}\hookrightarrow{\mathscr I}$ is a monomorphism with ${\mathscr I}$ injective in $\text{QCoh}(X)$. Hence $\text{QCoh}^G(X)$ has enough injectives. 
At first I was confused about this, since I rather expected a left adjoint of the forgetful functor, but a look at the case $X=\text{pt}$ helps: rational $G$-representations are comodules over the coalgebra $k[G]$, and in the category of comodules over a $k$-coalgebra $C$, the forgetful functor to $k\text{-Mod}$ has the cofree-comodule-functor $-\otimes_k C$ as its right adjoint.
Unfortunately, I cannot say anything about the second question, but maybe someone else does?
A: For the record, this is a skecth of what I have found out.
Let $\rho \colon G \to G/P$. It is easy to see that $\Gamma(U,\mathcal O_P(\lambda)) \cong \{f \colon \rho^{-1}(U) \to V \text{ such that } f(gp)= p^{-1} \cdot f(g) \text{ for all } g \in G, p \in P\}$.
Take $X \in \mathfrak g$ and consider it as a left invariant vector field on $G$. Then $X$ acts on $\Gamma(U,\mathcal O_P(\lambda))$, and we get an endomorphism of sheaf $\mathcal O_P(\lambda) \to \mathcal O_P(\lambda)$. Since $H^r(G/P,-)$ is a functor, we get an endomorphism $H^r(G/P,\mathcal O_P(\lambda)) \to H^r(G/P,\mathcal O_P(\lambda))$ (by the way, this space is finite dimensional by Cartan-Serre theorem). Since $G$ is simply connected, this action integrates to the action of $G$ on $H^r(G/P,\mathcal O_P(\lambda))$. This construction is from J. L. Taylor: Several complex variables with connections to algebraic geometry and Lie groups.
Alternatives to this and the answer of Hanno are following:
Use isomorphism between sheaf cohomology of $\mathcal O_P(\lambda)$ and Dolbeault cohomology of $G/P$. (R. Zierau: Representations in Dolbeault cohomology; A. W. Knapp: Introduction to representations in analytic cohomology)
Or use Čech cohomology of $\mathcal O_P(\lambda)$, which is, on a paracompact space, same as derived-functor cohomology. (D. N. Akhiezer: Lie group actions in complex analysis)
A: I want to expand on point 2 of Hanno, this is something I've been thinking of lately (in a simpler context). See my answer in Semi-linear endomorphism of sheaf of $k$-vector spaces induces a semi-linear endomorphism on cohomology, the approach is the same.
We can verify that the functor $f:\mathrm{QCoh}^G(X)\to \mathrm{QCoh}(X)$ sends injective objects to $\Gamma(U,-)$-acyclic objects by passing through Cech cohomology of presheaves. The forgetful functor $i$ to presheaves sends injectives to injectives, so if $I$ is injective, $i(I)$ is acyclic for Cech cohomology (defined as a derived functor of the 0-th term of the Cech-complex). But because the cohomology of the Cech complex coincides with the first one (this is the crucial nontrivial point ; it works for sheaves of abelian groups (done in Tamme, Introduction to Etale Cohomology) and for sheaves of $\mathcal{O}_X$-modules Tag01EN, and I think it works here, but I have to check it), the Cech complex for $i(I)$ is exact ; but this complex, as an abelian group, is the same as the one for $if(I)$, so that $if(I)$ also is acyclic for Cech cohomology. Now the Cech-to-derived spectral sequence gives that $f(I)$ is $\Gamma(X,-)$-acyclic.
