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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.

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    $\begingroup$ This certainly is true: en.m.wikipedia.org/wiki/Restricted_representation. I believe a proof can be found somewhere in Knapp’s book on semisimple Lie algebras, though I don’t have a copy on me. $\endgroup$ Jan 8, 2020 at 9:57
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    $\begingroup$ As far as I remember, the point is that $SO(n-1)$ is a (connected component) of fixed point group of an involution on $SO(n)$ (so the quotient is spherical). $\endgroup$ Jan 8, 2020 at 12:56
  • $\begingroup$ @VictorPetrov, that is so; see @‌FriedrichKnop's answer. $\endgroup$
    – LSpice
    Jan 8, 2020 at 18:45
  • $\begingroup$ It need to specify what is SO(n) (over what field, and defined by what quadratic forms). Anyway, there are two Annals papers on this question. arxiv.org/abs/0903.1413 annals.math.princeton.edu/wp-content/uploads/… $\endgroup$
    – Q. Zhang
    Jan 12, 2020 at 19:23

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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.

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    $\begingroup$ I voted to close as not a research-level question, but this answer is so good it makes me re-consider. $\endgroup$
    – LSpice
    Jan 8, 2020 at 17:15

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