rt.representation theory - Can an admissible SO(n) representation contain an SO(n-1) representation with infinite multiplicity?

Funnily enough, I wrote a paper about this question a few years ago.

The takeaway is that there is a geometric method for understanding when such a restriction is admissible. For many pairs of groups it never is, but I think Matt found the only examples of the form SO(n) and SO(n-1). In general, each finite dimensional representation of SO(n) comes from quantizing a coadjoint orbit, and you want only finitely many of the coadjoint orbits that lie in the image of the moment map on the contangent bundle of the sphere $T^*S^{n-1}$. In particular, $T^*S^1 to mathfrak{so}_2^*$ is surjective since $S^1$ is a regular action, and $T^*S^2 to mathfrak{so}_3^*$ is surjective since the adjoint representation is covered by the orbit of any line.