Exercise 3, § 9

Let G be a Lie group, H a compact subgroup of G. Assume that G admits a faithful (finite dimensional) linear representation.

a) Let $\Theta_H(G)$ be the subalgebra of $\mathscr{C}(H;\mathbf{R})$ consisting of the restrictions to H of the (continuous) representative functions on G. Show that $\Theta_H(G)$ is dense in $\mathscr{C}(H;\mathbf{R})$ for the topology of uniform convergence.

b) Let $f\in \Theta_H(G)$. Show that there exists a representation $\sigma$ of G on a finite dimensional real vector space such that a coefficient of the representation $\sigma |H$ is not orthogonal to $f($Spectral Theory, in preparation).

c) Let $\rho : H\rightarrow \mathbf{G}\mathbf{L}(V)$ be a representation of H on a finite dimensional real vector space. Show that there exists a finite dimensional real vector space W, a representation $\sigma : G\rightarrow \mathbf{G}\mathbf{L}(W)$ and an injective homomorphism $u: V\rightarrow W$, such that $u(\rho (h)v) =\sigma (h)u(v)$ for $h\in H,v\in V$.

(Reduce to the case in which $\rho$ is irreducible, and use $b$)$.)$