Proposition 54

Tag 0ABW, permanent. Cite it as [Bourbaki, Tag 0ABW].

Let G be a Lie group of finite dimension n, ω a left invariant differential form of degree n on G and φ an endomorphism of G. Then

\phi^*(\omega) = (\det L(\phi)) \omega.

We write $L(\phi) = u$ , $\omega_e = f$ and $\phi^*(\omega)_e = g$ . For all $x_1, \ldots, x_n$ in $L(G)$ ,

g(x_1, \ldots, x_n) = f(ux_1, \ldots, ux_n) = (\det u)f(x_1, \ldots, x_n)

and hence $\phi^*(\omega)_e = \det L(\phi) \cdot \omega_e$ . On the other hand, if $g \in G$ ,

\phi \circ \gamma(g) = \gamma(\phi(g)) \circ \phi

and hence $\gamma(g)^*\phi^*(\omega) = \phi^*(\omega)$ . Thus $\phi^*(\omega)$ is left invariant, whence the proposition.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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