Let $L\to X$ be a complex line bundle globally generated by sections. Suppose that there is action of a lie group $G$ on $\Gamma(X, L)$.
Is there necessarily an equivariant structure on $L$?
Here is my try at trying to get one:
Let $x \in X$, and let $v \in L_x$, and let $s$ be a global section through $v$. It makes sense to set $gv=(g*s)(g^{-1}x)$:
(after all if one already had an equivariant structure on $L$ then the action of $g$ on a section $s$, namely $g*s$, is supposed to be given by $gs(gx)=g*s$).
What has to be checked to see if this gives a well defined action on the total space of $L$ is that if $t$ is another section s.t. $t(x)=v$, then $(g*t)(g^{-1}x)=gv=(g*s)(g^{-1}x)$.
Is this true for some relatively trivial assumptions on $X$, $G$ and $L$?(I would also be happy for easy counter examples).