Let $G$ be a reductive group, $X$ be a projective variety and $\mathcal L$ an ample $G$ equivariant line bundle on $X$. Then by a descent lemma of Kempf (see Narasimhan, M.S., and Drezet, J.-M.. "Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques." Inventiones mathmaticae-1989) the line bundle $\mathcal L$ descends to the quotient $X//G$ iff for any $x \in X^{ss}$ the stabilizer $G_x$ acts trivially on the fiber $\mathcal L_x$.

Now my question is suppose we have a $G$-equivariant line bundle $\mathcal L$ which descends to the quotient $X//G$, is the descent unique ? If not is it unique if $G$ is finite say ?

equivariantbundles. That depends on the characters of the invertible functions on $X$. $\endgroup$