  The results in this paper provide a comparison between the $K$-structure of
unipotent representations and regular sections of bundles on nilpotent orbits.
Precisely, let $\widetilde{G_0} =\widetilde{Spin}(a,b)$ with $a+b=2n$, the
nonlinear double cover of $Spin(a,b)$, and let $\widetilde{K}=Spin(a, \mathbb
C)\times Spin(b, \mathbb C)$ be the complexification of the maximal compact
subgroup of $\widetilde{G_0}$. We consider the nilpotent orbit $\mathcal O_c$
parametrized by $[3 \ 2^{2k} \ 1^{2n-4k-3}]$ with $k>0$. We provide a list of
unipotent representations that are genuine, and prove that the list is complete
using the coherent continuation representation. Separately we compute
$\widetilde{K}$-spectra of the regular functions on certain real forms
$\mathcal O$ of $\mathcal O_c$ transforming according to appropriate characters
$\psi$ under $C_{\widetilde{K}}(\mathcal O)$, and then match them with the
$\widetilde{K}$-types of the genuine unipotent representations. The results
provide instances for the orbit philosophy.
