Projection and representation

Question

Suppose that $\rho : G \longrightarrow U_n(\mathbb C)$ is an irreducible representation of group $G$. Suppose that $P$ is a projection of $\mathbb C^n$ into a subspace of small codimension (i.e. of codimension $\varepsilon n$ for some small $\varepsilon$). Can you prove that $\mathbb{E}_{a} \rho(a)^{-1} P \rho(a)$ is close to identity? I mean is it true that:

$$|| \mathbb{E}_{a} \rho(a)^{-1} P \rho(a) - I ||=O(\varepsilon)$$

Answer 1

Assuming $G$ is compact, the answer is yes. The average of $\rho(a)^{-1}P\rho(a)$ over $a$ (this is where I use compactness) will be an operator that commutes with each $\rho(b)$. So by Schur's lemma this average is a multiple of the identity. To find out what multiple, take the trace; this tells us the average is ${\rm tr}(P)I$ where ${\rm tr}$ is the normalized trace on $M_n({\mathbb C})$, i.e. ${\rm tr}(I_n)=1$.