Im currently reading Jantzens "Representations of Algebraic Groups" 2nd Edition. On page 203 he explains how a $G$-linearized sheaf $\mathcal L$ induces a $G$-module structure on the cohomolgy groups $H^i(G/B, \mathcal L)$, for $G$ a reduced algebraic group and $B$ its Borel subgroup.
Then he claims that this implies that Serre Duality, as it is explained in Hartshorne III 7.2, is compatible with the $G$-action.
But why are these actions compatible with Duality?
At first I thought of interpreting the global section functor as a functor from the category of $G$-linearized sheaves to the category of $G$-modules and the higher cohomology groups as its right derived functors. This approach would enable me to define a $G$-structure on the $Ext$ groups via $Ext^i(\mathcal F,\mathcal G)\cong H^i(\mathcal O_{G/B}, \mathcal F^\vee \otimes \mathcal G)$ and hopefully use the naturality of the construction of the isomorphims $Ext^i(\mathcal F,\omega_{G/B}^\circ) \to H^{n-i}(G/B, \mathcal F)^*$ [Hartshorne III. 7.6a] in order to get the Serre duality for $G$-modules.
But now I am not really sure if this approach works. Does the category of $G$-linearized sheaves over $G/B$ even have enough injectives and if yes, are the resulting cohomology groups the same as the "original" groups? Furthermore, can I even use the same proof as in [HS III. 7.6a] to show that $Ext^i(-, {\omega_{G/B}}^\circ)$ is coeffaceable, or does the additional $G$-module structure hinder this?