Suppose that $G$ is a compact group and $C(G)$ is the space of continuous functions on $G$. For $f_1$, $f_2\in C(G)$, define the convolution by $$(f_1*f_2)(g)=\int_Gf_1(gh^{-1})f_2(h)\mathrm{d}h=\int_Gf_1(h)f_2(h^{-1}g)\mathrm{d}h.$$ I've verified the identity of the last two terms by the variable change $h\to h^{-1}g$, and I've shown that $C(G)$ is a ring with respect to convolution without unit by proving this operation is associate.
Now I have no idea to show that the space $\mathcal{M}_\pi$ of matrix coefficients of $\pi$ is a 2-side ideal in $C(G)$, where $\pi$ is an irreducible representation. After demonstrating this fact, I hope to explain how it implies Schur orthogonality.
Schur orthogonality. Suppose that $(\pi_1,V_1)$ and $(\pi_2,V_2)$ are irreducible representations of the compact group $G$. Either every matrix coefficient of $\pi_1$ is orthogonal in $L^2(G)$ to every matrix coefficient of $\pi_2$ or the representations are isomorphic.
Could you help me and give me some details?