The cohomology of a Borel-character line bundle is a rational G-module

For $g\in G$ let $a_g:X\to X$ be $x\mapsto gx$. The left action on the total bundle gives ${\mathrm{\Phi }}_g:{\mathcal{L}}_{\lambda }\to a_g^{\ast }{\mathcal{L}}_{\lambda }$, whose fibre map at $x$ sends $v\in ({\mathcal{L}}_{\lambda }{)}_x$ to $g\cdot v\in ({\mathcal{L}}_{\lambda }{)}_{gx}$. These maps are algebraic in $g$ and satisfy ${\mathrm{\Phi }}_{gh}=a_h^{\ast }{\mathrm{\Phi }}_g\circ {\mathrm{\Phi }}_h$: at $x$ the composite first applies $h$ to the fibre over $x$, then $g$ to the fibre over $hx$. Define ${\rho }_i(g):=H^i(a_{g^{-1}}^{\ast }{\mathrm{\Phi }}_g)\circ a_{g^{-1}}^{\ast }:H^i(X,{\mathcal{L}}_{\lambda })⟶H^i(X,{\mathcal{L}}_{\lambda })\mathrm{.}$ Here $a_{g^{-1}}^{\ast }$ first pulls cohomology to $H^i(X,a_{g^{-1}}^{\ast }{\mathcal{L}}_{\lambda })$, and the pulled-back bundle map $a_{g^{-1}}^{\ast }{\mathrm{\Phi }}_g:a_{g^{-1}}^{\ast }{\mathcal{L}}_{\lambda }\to {\mathcal{L}}_{\lambda }$ then returns to the original cohomology group. The cocycle gives ${\rho }_i(gh)={\rho }_i(g){\rho }_i(h)$ and ${\rho }_i(e)=\mathrm{i}\mathrm{d}$, so these are linear actions. They are natural in $\lambda$ because the bundle maps are.

The groups $V_i:=H^i(X,{\mathcal{L}}_{\lambda })$ are finite-dimensional: $X$ is smooth projective, hence proper over $\mathbb{C}$ by [F3], and ${\mathcal{L}}_{\lambda }$ is coherent by [F2], so [F4] applies.

On degree zero, the formula in step 1.1 sends a section to $x\mapsto g\cdot s(g^{-1}x)$, which corresponds by [F1] to left translation $(g\cdot f)(g^{\mathrm{\prime }})=f(g^{-1}{g}^{\mathrm{\prime }})$. If $\varphi :{\mathcal{L}}_{\lambda }\to {\mathcal{L}}_{\mu }$ is $G$-equivariant, its compatibility with the total-space action makes the induced family maps commute with $\varphi$; by functoriality [F4], every $H^i(\varphi )$ intertwines the actions ${\rho }_i$.

To prove rationality, put $R=\mathcal{O}(G)$, let $q:G\times X\to X$ be projection and define the algebraic automorphism $A:G\times X\to G\times X$ by $A(g,x)=(g,g^{-1}x)$. The product cover $G\times U_I$ and [F5] compute $H^i(G\times X,q^{\ast }{\mathcal{L}}_{\lambda })$ by the Cech complex whose terms, by [F6], are $R{\otimes }_{\mathbb{C}}\mathrm{\Gamma }(U_I,{\mathcal{L}}_{\lambda })$. Its differentials are $1_R\otimes d$, so exactness of tensoring over the field $\mathbb{C}$ identifies this cohomology with $R{\otimes }_{\mathbb{C}}{V}_i$. The left action on the line bundle gives an algebraic isomorphism $A^{\ast }{q}^{\ast }{\mathcal{L}}_{\lambda }\to q^{\ast }{\mathcal{L}}_{\lambda }$ over $G\times X$, sending the fibre over $(g,x)$ by the action of $g$ from $({\mathcal{L}}_{\lambda }{)}_{g^{-1}x}$ to $({\mathcal{L}}_{\lambda }{)}_x$. Pullback by $A$ followed by this isomorphism induces an $R$-linear endomorphism of $R\otimes V_i$. Under evaluation at each $g\in G$, the same product Cech computation identifies its fibre map with ${\rho }_i(g)$ of step 1.1. Thus the matrix entries of ${\rho }_i$ are regular functions on $G$; the action law makes ${\rho }_i:G\to \mathrm{GL}(V_i)$ an algebraic group homomorphism.

The identifications ${\mathcal{L}}_{\lambda }\otimes {\mathcal{L}}_{\mu }\cong {\mathcal{L}}_{\lambda +\mu }$ and ${\mathcal{L}}_{\lambda }^{\vee }\cong {\mathcal{L}}_{-\lambda }$ are morphisms of equivariant bundles by [F2], so the induced cohomology maps intertwine the actions from step 1.1. Each algebraic representation of step 2.2 differentiates to a $\mathfrak{g}$-module, and the tensor/dual identifications remain equivariant for both actions.