us to define the projection $P(\lambda)$ on $\Lambda_q(\lambda)$ by $$P(\lambda)v={1\over p}\sum_{g\in {\bf Z}_p}\exp[-2\pi i j/p]g\cdot v\no$$ Evidently $$[P(\lambda),\dstar d_q]=0\no$$ and so $P(\lambda)$ projects the space $\Lambda_q(\lambda)$ onto the space $\Lambda_q^{F}(\lambda)$. Finally this means that we obtain a formula for the degeneracy $\Gamma_n(q,p)$, namely $$\eqalign{\Gamma_n(q,p)&=
\tr\left(\left.P\right\vert_{\Lambda_q^{F}(\lambda)}\right)\cr
{}&={1\over p}\sum_{j=0}^{(p-1)}\exp[-2\pi i
j/p]\,\tr\left(\left.g\right\vert_{\Lambda_q^{F}(\lambda)}\right)\cr
}\no$$ To actually apply this formula we now add in the fact that $S^3$ is the group manifold for $SU(2)$. The Peter–Weyl theorem tells us, in this case where all representations are self-conjugate, that $$L^2(S^3)=L^2(SU(2))={\textstyle
\bigoplus\limits_{\mu}}\,\,c_{\mu}D_{\mu}=
{\textstyle \bigoplus\limits_{\mu}}\,\,D_\mu\otimes D_\mu\no$$ where $c_{\mu}$ measures the multiplicity of the representation $\mu$ which must therefore be $\dim D_\mu$. But Hodge theory gives us the alternative decomposition $$L^2(S^3)={\textstyle
\bigoplus\limits_{\lambda}}\,\,\Lambda_0(\lambda)\no$$ In addition the Casimir operator for $SU(2)$ is a multiple of the Laplacian and, if the representation label $\mu$ is taken to be the usual half-integer $j$, then we know that this Casimir has eigenvalues $j(j+1)$, and also that $\dim D_j=2j+1$. These facts identify the Laplacian $\Delta_0=\dstar d_0$ as four times the Casimir and identify $\Lambda_0(\lambda)$ as $\dim D_j$ copes of $D_j$. Thus if we set $n=2j$, so that $n$ is always integral, then we have the degeneracy formula $$\Gamma_n(0,p)={(n+1)\over p}\sum_{j=0}^{(p-1)}\exp[-2\pi i
j/p]\,\chi^{n/2}(2\pi j/p)\no$$ where $\chi^j(\theta)$ denotes 