Exercise 12, § 6

Let H be a connected closed subgroup of G and $\mathfrak{h}$ its Lie algebra.

$^{12}$The integers $k_i$ are in fact the exponents of $R(G,T)$. Cf. J. LERAY, Sur

l’homologie des groupes de Lie, des espaces homogènes et des espaces fibrés principaux, Colloque de Topologie de Bruxelles (1950), pp. 101-115. a) Let $\alpha$ be an element of $\mathfrak{h}^*$ invariant under H; show that there exists an element $\overline{\alpha}$ of $\mathfrak{g}^*$ invariant under H whose restriction to $\mathfrak{h}$ is equal to $\alpha$. Show that the element $d\overline{\alpha}\in$ Alt$^2(\mathfrak{g})$ is annihilated by $i(\eta )$ and $\theta (\eta )$ for all $\eta \in \mathfrak{h}$, and that its class in $H^2(G/H)$ does not depend on the choice of $\overline{\alpha}$. This defines a homomorphism $\varphi : H^1(H)\rightarrow H^2(G/H)$.

b) Assume from now on that G is semi-simple. Prove that $\varphi$ is an isomorphism. c) Prove that H is semi-simple if and only if $H^2(G/H) = 0$.

d) Without assuming that H is connected, define an isomorphism $(\mathfrak{h}^*)^H\rightarrow$ $H^2(G/H)$ (apply $b$) to $H_0)$.