# Orthogonality of $L^2(G)$ for a compact group

I am having trouble with the following:

Let $$G$$ be a compact group with Haar measure $$\mu$$ and $$(\rho,\mathbb{C}^n)$$ be an irreducible representation of $$G$$. Naturally, we can suppose that $$\rho$$ is unitary by Weyl's trick. Fix a $$\rho(G)$$-invariant inner product $$\langle-,-\rangle$$ on $$\mathbb{C}^n$$.

Suppose $$f\in L^2(G:\mathbb{C}^n)$$ is of the form $$f(g)=\rho(g)v$$ for some $$v\in \mathbb{C}^n$$. Let $$h\in L^2(G:\mathbb{C}^n)$$. Does there exist $$w\in \mathbb{C}^n$$ such that $$\int_G \langle f(g),h(g)\rangle d\mu=\int_G \langle f(g),\rho(g)w\rangle d\mu$$?

I believe the answer should hold by Peter-Weyl, but I am struggling to do the relevant calculations. Could someone please verify?

Thanks!

• I will try expressing $h$ as a (possibly infinite) sum of irreducible representations and then use the orthogonality relations between different irreducible representations to cancel all of them except $\rho$. – Adrián González-Pérez Oct 21 '19 at 11:42