Let $k$ be an algebraically closed field of characteristic $0$, $G/k$ be a linear algebraic group and $C/k$ be a curve. Let $F$ on $C$ be a $G$-bundle, $X/k$ be a projective $G$-variety and $L$ on $X$ be a $G$-line bundle.
Let $\mathscr{X}=F \times_{G} X$ (with structure morphism $\pi: \mathscr{X} \longrightarrow C$) and $\mathscr{L}=F \times_{G} L$ (which is a line bundle on $\mathscr{X}$). Show that $\pi_* \mathscr{L} = F \times_{G} H^0(X,L)$.
I believe I can prove this by writing down local things and gluing them by cocycle conditions. But I wonder if there is a smarter proof (or where could I find this in the literature, so that I can just cite it without proving it by myself).
