Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V_{ss}$ is a variety of semi-stable points). Let $E$ be a vector bundle on $X$ with a $G$-action commuting with the $G$-action on $X$. I believe that if $G$ acts effectively then there is a quotient vector bundle $E/G$ on $X/G$. Is it true? Could you give me references? Also, does all line bundles on $X/G$ arise this way?

I put my comment as an answer: the necessary and sufficient condition for $E$ to be the pull back of a vector bundle on $X/G$ is that the stabilizer of any closed point $x$ with a closed orbit
acts trivially on $E$. This is a lemma of Kempf, well explained in §2 of Drézet-Narasimhan *Groupe de Picard des variétés de modules de faisceaux semi-stables sur les courbes algébriques*, Invent. Math. 97 (1989), 53-94.

Groupe de Picard des variétés de modules de faisceaux semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53-94. $\endgroup$ – abx Oct 2 '16 at 12:52