Proposition 56

Let $G$ be a Lie group of finite dimension $n$, $H$ a $p$-dimensional Lie subgroup and $X$ the Lie homogeneous space $G/H$. Suppose that

for all $h \in H$. Then: (i) The differential forms of degree $n - p$ on $X$ which are invariant under $G$ are analytic. (ii) The vector space of these forms is of dimension 1. (iii) If $\omega$ is such a non-zero form and $K$ is locally compact, $\operatorname{mod}(\omega)_\mu$ is a non-zero measure on $X$ which is invariant under $G$.

By § 1, no. 8, Examples, $\operatorname{Alt}^{n-p}(TX, K)$ is an analytic vector $G$-bundle. Let $x_0$ be the canonical image of $e$ in $X$; its stabilizer is $H$. The fibre of $\operatorname{Alt}^{n-p}(TX, K)$ at $x_0$ is $\bigwedge^{n-p} T_{x_0}(X)^*$ and $T_{x_0}(X)$ is canonically identified with $L(G)/L(H)$. If $h \in H$, the automorphism $\tau_h$ of $X$ defined by $h$ is derived when passing to the quotient from the automorphism $g \mapsto hgh^{-1}$ of $G$. Hence the automorphism $T_{x_0}(\tau_h)$ is derived when passing to the quotient from $\operatorname{Ad}_{L(G)}(h)$. As

the hypothesis implies that $\det T_{x_0}(h) = 1$. Thus, every element of $\bigwedge^{n-p} T_{x_0}(X)^*$ is invariant under $H$. Then (i) and (ii) follow from § 1, no. 8, Corollary 1 to Proposition 17 and (iii) is obvious.

The existence of a non-zero positive measure on $X$ invariant under $G$ follows from Integration, Chapter VII, § 2, Corollary 2 to Theorem 3, for the hypothesis of Proposition 56 implies $\Delta_G|H = \Delta_H$ (Corollary to Proposition 55).