Exercise 7, § 9

Let G be a Lie group, H a compact subgroup of G, and $\rho$ a unitary representation of H on a real Hilbert space W. Denote by X the manifold $G\times^HW$ (no. 3).

a) Show that there exists a Hilbert space V, a unitary representation $\sigma$ of G on V and an (analytic) embedding $\varphi : X\rightarrow V$ such that $\varphi (gx) =\sigma (g)\varphi (x)$ for $g\in G,x\in X$ (use Exerc. 4 and Exerc. 6).

b) Prove that, if W is finite dimensional and if G admits a faithful finite dimensional linear representation, then V can be chosen to be finite dimensional (use Exerc. 3 and Exerc. 5).