Exercise 5, § 9

Let G be a Lie group, H a compact subgroup of G. Assume that there exists a faithful linear representation $\rho : G\rightarrow \mathbf{G}\mathbf{L}(V)$ of G on a finite dimensional real vector space.

a) Show that there exists a representation $\sigma$ of $\mathbf{G}\mathbf{L}(V)$ on a finite dimensional real vector space W, a separating positive quadratic form $q$ on V and a vector $w$ in W such that $\rho (H) =\mathbf{O}(q)\cap F_w$, where $F_w$ is the fixer of $w$ in $\mathbf{G}\mathbf{L}(V)$ (choose a quadratic form $q$ invariant under H and a representation of $\mathbf{O}(q)$ such that $\rho (H)$ is the fixer of a point (Cor. 2 of no. 2), then apply Exerc. 3 $c$)$)$.

b) Deduce from a) that there exists a finite dimensional real vector space E, a representation of G on E and a vector $e\in E$ with fixer H (take $E = W\oplus Q$, where Q is the space of quadratic forms on V, and $e= (w, q)$).