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Let $P \rightarrow X$ be a principal $G$-bundle, $\rho: G\rightarrow GL(V)$ and $\sigma: G\rightarrow GL(W)$ be two finite dimensional linear representations of $G$. Let $E=P\times_\rho V$ and $F=P\times_\sigma W$. By using transformation functions, it's easy to see that $E\oplus F\cong P\times_{\rho\oplus\sigma} (V\oplus W)$ and that $E\otimes F\cong P\times_{\rho\otimes\sigma} (V\otimes W)$.

My question is: Consider the left action of $G$ on the set $P\times V\times W$ below: $$a\cdot (u, v, w)=(ua^{-1}, \rho(a)(v), \sigma(a)(w)), $$ then what is the bundle structure of the quotient of $P\times V\times W$ by the above $G$-action?

I think it should be one of the above: $E\oplus F$ or $E\otimes F$, but I can not prove either.

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I think I have known the answer myself, it is $E\oplus F$, since the left action of $G$ on $P\times V\times W$ is the same as the left action of $G$ on $P\times (V\oplus W)$ via the representation $\rho\oplus\sigma$, thus coincides with the bundle $P\times_{\rho\oplus\sigma} (V\oplus W)\cong E\oplus F$.

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