A [paper][1] 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][2] 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][3] play out when $SO(4)$ acts on bundles of divergenceless vector fields and spinor-fields $S^3$?


  [1]: http://arxiv.org/abs/0909.4559
  [2]: http://math.stackexchange.com/questions/466113/the-spectrum-and-determinant-of-the-laplacian-on-s3
  [3]: https://terrytao.wordpress.com/2011/01/23/the-peter-weyl-theorem-and-non-abelian-fourier-analysis-on-compact-groups/