Proposition 2

Let H be a (real) Lie group of dimension $m,\mathfrak{h}$ its Lie algebra. Let $\omega_H$ be a right-invariant differential form of degree $m$ on H, and let $\omega_{\mathfrak{h}}$ be the translation-invariant differential form on $\mathfrak{h}$, of degree $m$, that coincides with $\omega_H(e)$ at the origin. We have

(exp$_H$)$^*\omega_H=\lambda_{\mathfrak{h}}\omega_{\mathfrak{h}}$ (10)

where $\lambda_{\mathfrak{h}}$ is the Ad(H)-invariant function on $\mathfrak{h}$ such that

$\lambda_{\mathfrak{h}}(x) =$ det $\sum(p+$ 1)!1 (ad $x$)$^p$ for $x\in \mathfrak{h}$.

$p\geq 0$

Let $x, x_1, . . . , x_m$ be elements of $\mathfrak{h}$. We have

(exp$^*\omega_H$)$_x(x_1, . . . , x_m) = (\omega_H$(exp $x$))$(T_x$(exp)($x_1$)$, . . . ,T_x$(exp)($x_m$)). Denote by $\varpi (x) :\mathfrak{h}\rightarrow \mathfrak{h}$ the right differential of the exponential at $x$ (Chap. III, §3, no. 17, Def. 8); by definition,

$T_x$(exp)($y$).(exp $x$)$^{-1}=\varpi (x).y$ for all $y\in \mathfrak{h}$.

The form $\omega_H$ being right invariant, we obtain

$(\omega_H$(exp $x$))$(T_x$(exp)($x_1$)$, . . . ,T_x$(exp)($x_m$))

$=\omega_H(e)(\varpi (x).x_1, . . . , \varpi (x).x_m) =$ (det $\varpi (x)$)$\omega_{\mathfrak{h}}(x_1, . . . , x_m)$;

thus, exp$^*\omega_H=\lambda_{\mathfrak{h}}\omega_{\mathfrak{h}}$, with $\lambda_{\mathfrak{h}}(x) =$ det $\varpi (x) =$ det$\frac{exp ad x-1}{ad x}$ (Chap. III, §6, no. 4, Prop. 12).

Let $h\in H$; since Ad $h$ is an automorphism of $\mathfrak{h}$, we have

ad ((Ad $h$)$(x)$) $=$ Ad $h\circ$ Ad$x\circ$ (Ad $h$)$^{-1}$,

so $\lambda_{\mathfrak{h}}$((Ad $h$)$(x)$) $=\lambda_{\mathfrak{h}}(x)$. Thus, the function $\lambda_{\mathfrak{h}}$ is invariant under Ad(H); this completes the proof of the proposition.