Spectra of the Laplacian operator on the spherical space-form Let $S^3/\Gamma$ be a spherical space form where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $S^3$.
If $\Gamma$ is trivial, it is well-known that the spectra of the Laplacian operator on $S^3$ are $k(k+2)$ for $k \ge 1$. Is there any reference for the spectra of the general $S^3/\Gamma$?
 A: In addition of what you mentioned, the multiplicity of $k(k+2)$ in the Laplace spectrum of $S^3$ is given by $\dim H_k$, where $H_k$ is the space of harmonic homogeneous complex-valued polynomials of degree 2 in four variables.
One can easily check that $\dim H_k=\binom{k+3}{3}-\binom{k+1}{3}$.
Now, the Laplace spectrum of $S^3/\Gamma$ (note that $\Gamma\subset SO(4)$ since $\Gamma$ acts free on $S^3$ by assumption) has eigenvalues $k(k+2)$ with multiplicity $\dim H_k^\Gamma$.
Here, $H_k$ is considered as a representation of $SO(4)$ and $H_k^\Gamma$ the $\Gamma$-invariant elements in $H_K$.
Note that it may occur that $\dim H_k^\Gamma=0$, in which case $k(k+2)$ is not longer an eigenvalue of $S^3/\Gamma$.
If you wish to compute explicitely the Laplace spectrum of $S^3/\Gamma$ (i.e. to know $\{\dim H_k^\Gamma:k\geq0\}$), you may study the generating function
$$
F_{\Gamma}(z):= \sum_{k\geq0} \dim H_k^{\Gamma} z^k. 
$$
Ikeda and Yamamoto [Osaka J. Math 16 (1979), 447--469] proved that
$$
F_{\Gamma}(z) = \frac{1-z^2}{|\Gamma|} \sum_{\gamma\in\Gamma} \prod_{\lambda\in Spec(\gamma)}\frac{1}{1-\lambda z}.
$$
This was also done by Wolf [Result. Math. 40 (2001), 321--338] by applying Molien's formula.
