The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$ The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia article. Am I correct in reading from this that any given representation of $SO(n-1)$ can appear in a representation of $SO(n)$ with multiplicity at most $1$? Also what is the name for the corresponding homogeneous space, and does it have any interesting geometric properties?
Also, I am interested in the same questions for $SO(n)/O(n-1)$ and $O(n)/O(n-1)$.
 A: Yes, the branching is multiplicity free. See e.g. Theorem 8.1.3 and Theorem 8.1.4 in Symmetry, Representations, and Invariants by Nolan Wallach and Roe Goodman. The quotient space is a sphere $S^{n-1}$ which you can see for example by calculating the stabilizer of $e_n.$
For orthogonal groups you basically have extra $\mathbb{Z}_2$ action, so for example $SO(n)/O(n-1)$ comes from considering action of $SO(n)$ on $\mathbb{R}^n / \mathbb{Z}_2$ where the action of the $\mathbb{Z}_2$ is by multiplication by $-1.$ As for the branching rules in these cases ... it gets a bit complicated and it's hard to find any good reference. What happens here is that the branching is really a statement about Lie algebra representations (or equivalently about the simply connected covering groups $Spin(n)$) and for disconnected groups such as $O(n)$ one has to specify the action of all connected components. See e.g. https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group#Space_inversion_and_time_reversal for well studied examples.
