Pardon my ignorance, but I have a very basic question about the definition of a vector bundle from a principal G-bundle.
Let $G$ be a Lie group (I mainly care about the case $G=U(1)$). Let $P \rightarrow M$ be a principal bundle. Furthermore let $V$ be a vector space (over $\mathbb{R}$ or $\mathbb{C}$ - it shouldn't matter), and $\phi: G \rightarrow Gl(V)$ a representation of $G$. Then one can define a vector bundle over $M$, which topologically is given by $(P \times V)/G$ where $G$ acts by $$g.(p,v) = (g^{-1}p,\phi(g).v) .$$
My question is: Why take $g^{-1}$ in the first co-ordinate and not $g$? what would go wrong if we take g instead?
In the usual conventions, a principal $G$-bundle carries a canonical right action of $G$, and one often also has $G$ acting from the right on $P\times V$. It seems that you want to have $G$ acting from the left on $P\times V$, so your formula should read as $g\cdot (p,v)=(pg^{-1},\phi(g)\cdot v)$. The inversion is needed to switch between left and right actions, since otherwise multiplicativity does not work out.