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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?

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    $\begingroup$ No, this is false. You need that the stabilizer of a closed point $x$ acts trivially on $E_x$. For a precise statement, see §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. $\endgroup$ – abx Oct 2 '16 at 12:52
  • $\begingroup$ @abx Seems like you could just add that as an answer. $\endgroup$ – Ben Webster Oct 2 '16 at 13:24
  • $\begingroup$ @abx, thank you, and add this as an answer, please. $\endgroup$ – evgeny Oct 2 '16 at 13:44
  • $\begingroup$ related : mathoverflow.net/questions/194955/line-bundle-descends, where @abx gives the same answer $\endgroup$ – Niels Oct 2 '16 at 20:22
  • $\begingroup$ Related: MSE:1990550, there is a link to an article by Knop, Kraft, Vust with very clear explanation. $\endgroup$ – evgeny Oct 29 '16 at 20:07
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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.

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