Number of SO(3) orbits

Question

Let $D^l$ be a $n=2l+1$ dimensional irreducible unitary representation of $SO(3)$. Then $SO(3)$ acts on the sphere $S^{2n-1}\subseteq \Bbb{C}^n$. ($D^l(R)$ is unitary $\forall R \in$ $SO(3) )$. Into how many orbits will the sphere decompose?

Answer 1

Once $l>0$ there are infinitely many orbits just for dimension reasons: $SO(3)$ is three-dimensional, but the unit sphere in ${\bf C}^n$ has dimension $2n-1 = 4l+1 > 3$. Of course for $l=0$ the action is trivial, so again there are infinitely many orbits.

Likewise, once $l>1$ there are already infinitely many orbits for the action of $SO(3)$ on the unit sphere in ${\bf R}^n$, which has dimension $n-1 = 2l$. (For $l=1$ the action is transitive, and for $l=0$ the action is trivial but $S_0$ has only two points so there are two orbits.)