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I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections of the the vector bundle $V$. I've heard one can construct a vector bundle $\tilde{p}: \tilde{V} \to B$ such that $\Gamma(\tilde{V},B)$ is isomorphic to $\Gamma_G(V,M)$. How can such vector bundle be constructed? Does it have a special name or usual notation? (Maybe $V/G$)

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I suppose $V\to M$ is a $G$-equivariant vector bundle. Then $\bar V$ is defined as follows: the fiber $\bar V|_b$ over a point $b\in B$ is (canonically) equal to the space of $G$-invariant sections of $V|_{\pi^{-1}(b)}\to \pi^{-1}(b)$. The latter space has dimension equal to the rank of $V$.

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