I am trying to prove the following fact.


Let $V$ be a unitary irreducible representation of $SO(n)$. How to prove that, if we reduce $V$ as unitary irreducible representation with respect to SO(n-1) then each irreducible representation of $SO(n-1)$ occurs at most once in $V$.


Kindly share your thoughts.


Thank you.



Multiplicity of an irrep of SO(n-1) in SO(n)
The statement is true and well known. See e.g. Thms. 8.1.3 and 8.1.4 in Goodman-Wallach: Symmetry, representations and invariants, Springer GTM 255.


In fact, much more is known. Let, more generally, $H\subseteq G$ be a subgroup (everything is compact). Then it follows easily from Peter-Weyl that $G$-to-$H$ branching is multiplicity free if and only if $\mathbb C[G]$ is multiplicity free as a $G\times H$-module. This means that $G\times H/{\rm diag}\, H$ is a spherical $G\times H$-variety. There are (up to trivial modifications of the centers) only two series of indecomposable pairs $(H,G)$ with multiplicity free branching namely $U(n-1)\subset U(n)$ and $SO(n-1)\subset SO(n)$. This was known much earlier to Kostant but also follows easily from independent classifications classifications of Brion and Mikityuk.


The spaces $G\times H/H$ for $G$ unitary or orthogonal can also be defined for indefinite scalar products or over local fields. Then they are called Gross-Prasad spaces. Gross and Prasad stated a conjecture on the multiplicity freeness of these spaces which has attracted a lot of attention in recent years.


