Proposition 1

Let $\omega_G, \omega_{G/T}$ and $\omega_T$ be invariant differential forms on $G,G/T$ and T, respectively, of respective degrees $n, n-r$ and $r$. If $\omega_G=$ $\omega_{G/T}\cap \omega_T$, then

Clearly we can assume that $\omega_{G/T}$ and $\omega_T$ are non-zero; then the differential form $(u, t) \rightarrow \omega_{G/T}(u)\wedge \omega_T(t)$ on $(G/T)\times T$ is of degree $n$ and everywhere non-zero; hence there exists a numerical function $\delta$ on $(G/T)\times T$ such that

Observe now that, for $h\in G,u\in G/T,t\in$ T, we have $f(h.u, t) =$ (Int $h$)$f(u, t)$; since $\omega_G$ is invariant under inner automorphisms, it follows immediately that $\delta (h.u, t) =\delta (u, t)$, so $\delta (u, t) =\delta (\overline{e}, t)$.

Denote by $p:\mathfrak{g}\rightarrow \mathfrak{g}/\mathfrak{t}$ the quotient map and by $\varphi :\mathfrak{g}/\mathfrak{t}\rightarrow \mathfrak{g}$ the map defined by

$\varphi (p(X)) =$ (Ad $t^{-1}$)$X-X$ for $X\in \mathfrak{g}$; recall (§5, no. 4, Lemma 4) that the tangent map

takes $(z, tH)$ to $t(\varphi (z) +H)$ for $z\in \mathfrak{g}/\mathfrak{t}, H\in \mathfrak{t}$.

Let $z_1, . . . , z_{n-r}$ be elements of $\mathfrak{g}/\mathfrak{t},H_1, . . . , H_r$ elements of $\mathfrak{t}$. Then

$=\omega_G(t)(t\varphi (z_1), . . . , t\varphi (z_{n-r}), tH_1, . . . , tH_r)$ by the calculation of $T_{(\overline{e},t)}(f)$

$=\omega_G(e)(\varphi (z_1), . . . , \varphi (z_{n-r}), H_1, . . . , H_r)$ since $\omega_G$ is invariant

$=\omega_{G/T}(\overline{e})(p\varphi (z_1), . . . , p\varphi (z_{n-r})).\omega_T(e)(H_1, . . . , H_r)$ (no$.1$, Lemma 1)

= det($p\varphi$ )$\omega_{G/T}(\overline{e})(z_1, . . . , z_{n-r}).\omega_T(e)(H_1, . . . , H_r)$

since $\omega_T$ is invariant

so $f^*\omega_G(\overline{e}, t) =\delta_G(t)(\omega_{G/T}\wedge \omega_T)(\overline{e}, t)$; thus, $\delta (\overline{e}, t) =\delta_G(t)$, hence the proposition.

Give the manifolds $G,T$ and $G/T$ the orientations defined by the forms $\omega_G, \omega_T$ and $\omega_{G/T}$, respectively. These forms define invariant measures on $G,T$ and $G/T$ (Chap. III, §3, no. 16, Props. 55 and 56), also denoted by $\omega_G, \omega_T$ and $\omega_{G/T}$.