Exercise 11, § 6

Let $u$ be an automorphism of G of order two; denote by K the identity component of the set of fixed points of $u,\mathfrak{k}$ its Lie algebra, and X the symmetric space $G/K ($§1, Exerc. 8).

a) Show that every G-invariant differential form $\omega$ on X satisfies $d\omega = 0$ (observe that $u$ induces on Alt$^p(\mathfrak{g}/\mathfrak{k})$ the multiplication by $(-1)^p$).

b) Deduce that there is an isomorphism from the graded algebra $H(G/K)$ to the graded subalgebra of Alt($\mathfrak{g}/\mathfrak{k}$) consisting of the K-invariant elements.

c) Put $b_i(G/K) =$ dim$_{\mathbf{R}}H^i(G/K)$ for $i\geq 0$; for $k\in K$, denote by Ad$^-k$ the restriction of Ad $k$ to the eigenspace of $L(u)$ with eigenvalue $-1$. Let $dk$ be the Haar measure on K of total mass 1. Prove the formulas $b_i(G/K) =$ $\int_K$ Tr $\wedge \wedge^i$(Ad$^-k$)$dk$ and $\sum_{i\geq 0}b_i(G/K)X^i=\int_K$ det(1 + X Ad$^-k$)$dk$.

d) Prove that if, in addition, K is of maximum rank, the algebra $H(G/K)$ is zero in odd degrees (observe that $u=$ Int $k$, with $k\in K$).

e) Calculate the graded algebra $H(\mathbf{S}_n)$. Deduce that $\mathbf{S}_n$ admits a Lie group structure (compatible with its manifold structure) if and only if $n$ is equal to 1 or 3.