Let $L \to X$ be a $G$ equivariant line bundle over a projective variety $X$.
Then one has a $G$ action on the space of sections: a section $s: X \to L$ is sent to $g s (g x)$.
I don't understand how one can get an action of $G$ on $H^i(X,L)$ except for a few special cases. How can one get the action?
There are a few special cases I know where I can get an action by serre duality. For instance if $G$ is $SL_2$, $X=G/B=\mathbb{P^1}$, $L$ is any line bundle, say $\mathcal O(d)$ for some $d$ then we know that from the cech resolution, that $H^1(\mathcal O(-d))=\mathbb{C}^{d-1}$ for $d \geq 2$ and zero otherwize and the $G-$module structure comes using that $H^1(\mathcal (O(-d)))=H^0(\mathcal O(d) \otimes K_x)^*=H^0(\mathcal O(d-2))^*=Sym^{d-2}(\mathbb{C}^{2})$ which is an irreducible representation of highest weight $(d-2) \epsilon_1$.
But it would be nice if I could see how to get an action on the higher cohomology groups and conclude using something else that the action is the same as what you get using Serre duality.