I was writing up some notes on harmonic analysis and I thought of a question that 
I felt I should know the answer to but didn't, and I hope someone here can help me. 
Suppose I have a compact Riemannian manifold $M$ on which a compact Lie group $G$ 
acts isometrically and transitively---so you can think of $M$ as $G/K$ for some closed 
subgroup $K$ of $G$. Then the real Hilbert space $H = L^2(M, \mathbb{R})$ is an orthogonal representation space of $G$ and hence splits as an orthogonal direct sum of finite 
dimensional irreducible sub-representations. On the other hand, the Laplacian $L$ 
of $M$ is a self-adjoint operator on $H$, so $H$ is also the orthogonal direct sum 
of its eigenspaces---which are also finite dimensional. My question is, when do these 
two orthogonal decompositions of $H$ coincide? Put slightly differently, since $L$ 
commutes with the action of $G$, each eigenspace of $L$ is a finite dimensional 
subrepresentation of $H$ and so a direct sum of irreducibles, and I would like to
know conditions under which each eigenspace is in fact irreducible. For example, this
is true for the circle acting on itself and for $SO(3)$ acting on $S^2$ (where we
get the harmonic polynomials of various degrees). Is it perhaps always true for the 
case of a symmetric space? Of course a standard reference in addition to the answer 
would be most welcome.



When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?
The Peter-Weyl theorem tells you that $L^2(G)$ is isomorphic to $\bigoplus\_{\pi}\pi\otimes\pi^\*$ as $G\times G$ representation, where $\pi$ runs through all irreducible unitary representations.
It follows that
$$
L^2(G/K)\cong L^2(G)^K\cong\bigoplus\_\pi \pi\otimes(\pi^\*)^K.
$$
So, the first thing you absolutely need, is a multiplicity one property, which says that $\dim\pi^K\le 1$ for every $\pi$.
This is already a rare property, but known to be true for, say $G=SO(n)$ and $K=SO(n-1)$, see Zhelobenko's book for this.
But, the Laplacian may have the same eigenvalue on different representations.
For this you need highest weight theory (see for instance the book by Broecker and tom Dieck): Assume $G$ to be connected. The irreducible representations are parametrized by highest weights and the Laplace eigenvalue depends on the value of a quadratic form on the space of weights. 
So, in each case you need to identify those weights with $K$-invariants and consider the values of the quadratic form, which in the case of a simple group should be the Killing form. I guess that in the above cases it might actually be true.


