We have $$e^{i\lambda x}\cdot e^{i\mu x}=e^{i(\lambda+\mu) x}.\label{1}\tag{1}$$ 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 \eqref{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 like the above or estimates such as $$c_{\lambda,\mu}^\nu=0\text{ for }|\nu-(\lambda+\mu)|\geq C?$$
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1$\begingroup$ Such question are answered by the Steinberg multiplicity formula (a consequence of the Weyl character formula). 'Clebsch-Gordan' usually means the results of this formula applied to either $\mathrm{SO}(3)$ or $\mathrm{SU}(2)$. There, the answer is relatively simple, since $$\pi_\lambda\otimes\pi_\mu \simeq \pi_{\lambda+\mu}\oplus \pi_{\lambda+\mu-2}\oplus \pi_{\lambda+\mu-4}\oplus\cdots\oplus\pi_{|\lambda-\mu|}$$ for $\lambda,\mu\ge0$. $\endgroup$– Robert BryantCommented Dec 22, 2020 at 10:29
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$\begingroup$ @Professor Bryant I see. I will try to analyze the Steinberg formula and look for their vanishing properties. The formula looks like some sort of alternating sum, so the question of vanishing seems interesting... $\endgroup$– shrinklemmaCommented Dec 23, 2020 at 1:41
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$\begingroup$ Looking at this again, I realize that I should have specified that, in the case of $\mathrm{SU}(2)$, the weights $\lambda$, $\nu$, and $\mu$, etc. range over the nonnegative integers, while, in the case of $\mathrm{SO}(3)$, these weights range over the nonnegative even integers. Note that, in both cases, $\dim_\mathbb{C}V_\lambda = \lambda+1$, and, when $\lambda$ is even, $V_\lambda = \mathbb{C}\otimes W_\lambda$ where $W_\lambda$ is a real representation. $\endgroup$– Robert BryantCommented Jan 4, 2021 at 13:37
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