Proposition 3

Let $\omega_{\mathfrak{g}},\omega_{\mathfrak{t}},\omega_{G/T}$ be invariant differential forms on $\mathfrak{g},\mathfrak{t}$, $G/T$, respectively, of respective degrees $n, r, n-r$. If $\omega_{\mathfrak{g}}=\omega_{G/T}\cap \omega_{\mathfrak{t}}$, we have

where $\pi_{\mathfrak{g}}$ is the function on $\mathfrak{t}$ defined by $\pi_{\mathfrak{g}}(x) =\prod_{\alpha\in R(G,T)}\delta (\alpha )(x)$.

Denote by $\omega_G$ (resp. $\omega_T$) the invariant differential form of maximum degree on G (resp. T) that coincides with $\omega_{\mathfrak{g}}$ (resp. $\omega_{\mathfrak{t}}$) at the origin. Consider the commutative diagram

(Id,exp$_T$) exp$_G$.

$(G/T)\times T\longrightarrow^f$ G

In view of Prop. 1 of no. 2 and the relation exp$^*_T\omega_T=\omega_{\mathfrak{t}}$, we deduce the equality

$\psi^*$exp$^*_G\omega_G=\omega_{G/T}\wedge \delta_{\mathfrak{g}}\omega_{\mathfrak{t}}$.

By Prop. $2,\psi^*$exp$^*_G\omega_G= (\psi^*\lambda_{\mathfrak{g}})\psi^*\omega_{\mathfrak{g}}$. Since the function $\lambda_{\mathfrak{g}}$ is invariant under Ad(G), we have $(\psi^*\lambda_{\mathfrak{g}})(\overline{g}, x) =\lambda_{\mathfrak{g}}(x) =\frac{\delta_{\mathfrak{g}}(x)}{\pi_{\mathfrak{g}}(x)}$ for $\overline{g}\in G/T, x\in \mathfrak{t}$.

It follows that the forms $\psi^*\omega_G(\overline{g}, x)$ and $\omega_{G/T}(\overline{g})\wedge \pi_{\mathfrak{g}}(x)\omega_{\mathfrak{t}}(x)$ coincide where $\delta_{\mathfrak{g}}(x)$ is non-zero, that is on the dense open subset $(G/T)\times \mathfrak{t}_r$; thus, they are equal, hence the proposition.

Choose invariant differential forms $\omega_G$ on G and $\omega_T$ on T, of maximum degree, such that $|\omega_G|=dg$ and $|\omega_T|=dt$; denote by $\omega_{\mathfrak{g}}$ (resp. $\omega_{\mathfrak{t}}$) the translation-invariant differential form on $\mathfrak{g}$ (resp. $\mathfrak{t}$) that coincides with $\omega_G(e)$ (resp. $\omega_T(e)$) at the origin, and $dz$ (resp. $dx$) the Haar measure $|\omega_{\mathfrak{g}}|$ (resp. $|\omega_{\mathfrak{t}}|$). Reasoning as in no. 2, mutatis mutandis, gives the following proposition: