Integration on Lie groups ( In the proof of existence of the Haar volume form on $G$ ) I'm reading John Lee's Introduction to smooth manifold, p.410, Prop.16.10 and some question arises :


Why the underlined statement is true?
Perhaps, $cL_{g}^{*} \omega_G|_{e} = c\omega_{G}|_{g}$ is more correct?
Is there a point that I misunderstood? Where?
 A: That underlined portion consists of three equations, and it's not quite clear to me how your one proposed equation is intended to fit into those three. So altogether it's rather unclear to me what about those equations is unclear to you.
However (and although I might be entirely off the mark), your proposed equation gives me a hint regarding what you might have misunderstood, namely a confusion regarding pullback operators such as $L_g^*$. The equation you wrote does not make much sense. $L_g^*$ is a pullback operator, it does not take objects at $e$ to objects at $g$ as your equation would have it. Instead, $L_g^*$ takes objects at $e$ to objects at $g^{-1}$ because its dual $L_g$ takes objects at $g^{-1}$ to objects at $g g^{-1} = e$.
Looking at the final one of the three underlined equations, that equation does indeed make sense: $L^*_{g^{-1}}$ takes objects at $e$ to objects at $g$, because its dual $L_{g^{-1}}$ takes objects at $g$ to objects at $g^{-1}g=e$. And since $\omega_G$ is left invariant, $L^*_{g^{-1}}$ takes $\omega_G \mid_e$ to $\omega_G \mid_g$. So if that final equation is what you were asking about, perhaps it is all cleared up? Otherwise... I dunno.
