Action on sheaf cohomology in Bott-Borel-Weil theorem

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:

• 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?