I am trying to understand an argument in Guillemin and Sternberg's paper *Geometric Quantization and Multiplicities of Group Representations* (Inventiones, 1982). The argument (Proof of Theorem 3.2) seems to be based on the following fact:


**Lemma (I think).** *Let $G$ be a compact connected Lie group acting smoothly and freely on a smooth manifold $M$ and let $L$ be a $G$-equivariant complex line bundle on $M$. Then there is a complex line bundle $L\_G$ on $M/G$ such that $\pi^\*L\_G = L$, where $\pi : M \to M/G$ is the quotient map.*


They claim this result (without proof) in a more specific setting (where $M$ is the zero fibre of a moment map and $L$ is the pullback of a prequantum line bundle), but I don't see why it should hold.


More specifically, they define $L\_G$ by the sheaf of $G$-invariant sections of $L$, but I don't see why it is locally free of rank 1. It is easy to see that the lemma is equivalent to the following claim:


**Claim.** *For every $p \in M$ there is a $G$-invariant neighbourhood $U$ of $p$ in $M$ together with a non-vanishing $G$-invariant section $s : U \to L$.*



Quotient of line bundle by compact Lie group (after Guillemin-Sternberg)
Yes, the claim is true. It's a special case of a more general fact, that, in quite some generality, equivariant vector bundles are equivariantly locally trivial.


In your case, given $p$, there is a "slice" through $p$, a set $S\subset M$ containing $p$ such that $G\times S \to M$ is one-to-one and a homeomorphism onto an open subspace. By intersecting with a small enough nonequivariant neighborhood, we can assume that $L$ is trivial on $S$, and the action of $G$ then gives an equivariant trivialization of $L$ on $U = GS$. For a line bundle, an equivariant trivialization is equivalent to having a non-vanishing $G$-invariant section (assuming that the isotropy group, in this case the trivial group, acts trivially on the fiber at $p$).


Slices exist pretty generally, as shown by Palais. See, for example, the writeup at the [nLab](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/slice+theorem).


