I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository information on them.
It seems like one idea of them is if you have a manifold $X$ and a group action of $G$, then $X/G$ might be singular so you can't really have a vector bundle on it. A $V$-bundle is a vector bundle analog where your $V$-bundle is a vector bundle $E$ on $X$ with a family of unitary isomorphisms $$ \phi_x(g):E(x) \mapsto E(g\cdot x) $$ Where $E(x)$ is the fiber over $x\in X$. It also needs to satisfy a compatibility condition with the group action. So it's a vector bundle over $X$ but you identify fibers in an appropriate way.
What I'm really after is if $X/G$ is smooth, then does a $V$--bundle on $X/G$ actually give a vector bundle on $X/G$? The other direction is less important for me but would still be nice to know.
Thanks for the help!