Exercise 10, § 6

Let H be a closed subgroup of G; denote by $dh$ the Haar measure on H of total mass 1. Put $\chi (G/H) =\sum_{i\geq 0}(-1)^i$ dim$_{\mathbf{R}}H^i(G/H)$.

a) Prove the equality $\chi (G/H) =\int_H$ det(1$-$Ad$_{\mathfrak{g}/\mathfrak{h}}h$)$dh$ (use Algèbre, Chap. X, p. 41, Prop. 11 and Lemma 1 of Appendix II, no. 2).

b) Let $\pi_0(H)$ be the number of connected components of H. Show that

$\chi (G/H) = 0$ if $H_0$ is not of maximum rank

$\chi (G/H) =w(G)/w(H_0)\pi_0(H)$ if $H_0$ is of maximum rank.