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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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Branching laws for $SO(n)$

The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article: https://en.wikipedia.org/wiki/Restricted_representation#CITEREFMu …