A [paper](http://arxiv.org/abs/0909.4559) on supersymmetry in 3-dimensions uses results on the spectra of elliptic operators on $S^3$:



> 
> The eigenvalues of the vector Laplacian on divergenceless vector
>  fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell \in \mathbb{ Z}$.
> 
> 
> 


They came up with a similar statement for spinor fields on the 3-sphere



> 
> The eigenvalues of the Dirac operator $D$ is $\pm (\ell + \frac{1}{2})$ with degeneracy $\ell(\ell+1)$ with $\ell \in \mathbb{ Z}$.
> 
> 
> 


On [Math.StackExange](https://math.stackexchange.com/questions/466113/the-spectrum-and-determinant-of-the-laplacian-on-s3) it was suggested we can restrict the eigenspace decomposition the Laplacian on $SO(4)$ to $S^3$:
$$ L^2(SO(4)) = \bigoplus\_{\pi \in \mathrm{Irr}[SO(4)]} \pi \otimes \overline{\pi}$$
In our case, how [Peter-Weyl Theorem](https://terrytao.wordpress.com/2011/01/23/the-peter-weyl-theorem-and-non-abelian-fourier-analysis-on-compact-groups/) play out when $SO(4)$ acts on bundles of divergenceless vector fields and spinor-fields $S^3$?



the spectrum of the Laplacian and Dirac operator on $S^3$
Let $M=G/K$ with $G$ compact and let $(\tau,W\_\tau)$ be an irreducible representation of $K$, let $E\_\tau$ be the associated $G$-homogeneous vector bundle of $M$. 
Then, the space of $L^2$-sections of $E\_\tau$ decomposes as 
$$
L^2(E\_\tau) = \sum\_{\pi\in\widehat G} V\_\pi\otimes \operatorname{Hom}\_K(V\_\pi,W\_\tau).
$$


The proof follows by Peter-Weyl:
$\Gamma^\infty(E\_\tau)\simeq C^\infty(G;\tau):=\{f:G\to W\_\tau:f(gk)=\tau(k)^{-1}f(g)\}$. 
For each $\pi\in\widehat G$, we have $V\_\pi\otimes \operatorname{Hom}\_K(V\_\pi,W\_\tau) \to C^\infty(G;\tau)$ given by $v\otimes L \mapsto (x\mapsto L(\pi(x^{-1}).v))$. This map is $G$-equivariant. 


An other way, $C^\infty(G;\tau) = C^\infty(G,W\_\tau)^K \simeq (C^\infty(G)\otimes W\_\tau)^K$. Peter-Weyl theorem implies that 
$$
L^2(E\_\tau) = \sum\_{\pi\in\widehat G} (V\_\pi\otimes V\_\pi^\*\otimes W\_\tau)^K = \sum\_{\pi\in\widehat G} V\_\pi\otimes (V\_\pi^\*\otimes W\_\tau)^K = \sum\_{\pi\in\widehat G} V\_\pi\otimes\operatorname{Hom}\_K(V\_\pi,W\_\tau).
$$


When $M$ is symmetric (e.g. $M=S^3$) and $G$ semisimple, the Laplace operator acts as the Casimir element of $\mathfrak g$ (the Lie algebra of $G$). Furthermore, the square of the Dirac operator $D^2$ coincides with the Casimir element plus $1/8$ the scalar curvature. See Section 3.5 of T. Friedrich, "Dirac operators in Riemannian geometry". 


For example, let $S^{3}=\operatorname{Spin}(4)/\operatorname{Spin}(3)$, let $\tau$ be the spin representation, then $E\_\tau$ is the spinor bundle. $\tau$ has highest weight $\frac12 \varepsilon\_1$, thus $\operatorname{Hom}\_K(V\_\pi,W\_\tau)\neq0$ if and only if $\pi$ has highest weight $\Lambda = \frac12((2k+1)\varepsilon\_1\pm\varepsilon\_2)$. The casimir element acts on $\pi\_\Lambda$ as $\|\Lambda+\varepsilon\_1\|-\|\varepsilon\_1\|$ (see Wallach's book, Harmonic analysis on homogeneous vector bundles). Check that is equal to 
$$
\frac{(2k+1)(2k+5)+1}{4}
$$
The sectional curvature is 6, then $D^2$ acts by $(k+3/2)^2$. Finally, the eigenvalues of $D$ are $\pm(k+3/2)$.


