We have 
$$e^{i\lambda x}\cdot e^{i\mu x}=e^{i(\lambda+\mu) x}.\ (1)$$
More generally, consider the Clebsch–Gordan coefficients $c_{\lambda,\mu}^\nu$ defined by 
$$\pi_\lambda\otimes\pi_\mu=\sum_\nu c_{\lambda,\mu}^\nu \pi_{\nu}\ $$
where $\pi_\alpha$ stands for the irreducible representation of a compact Lie group of spectral parameter $\alpha$. On the circle group, because of (1), we have 
$$c_{\lambda,\mu}^\nu=0\text{ except when }\nu=\lambda+\mu.$$
My question is, in general, are there any simple vanishing results or estimates like the above?