  This paper provides a comparison between the $K$-structure of unipotent
representations and regular sections of bundles on nilpotent orbits for complex
groups of type $D$. Precisely, let $ G_ 0 =Spin(2n,\mathbb C)$ be the Spin
complex group viewed as a real group, and $K\cong G_0$ be the complexification
of the maximal compact subgroup of $G_0$. We compute $K$-spectra of the regular
functions on some small nilpotent orbits $\mathcal O$ transforming according to
characters $\psi$ of $C_{ K}(\mathcal O)$ trivial on the connected component of
the identity $C_{ K}(\mathcal O)^0$. We then match them with the ${K}$-types of
the genuine (i.e. representations which do not factor to $SO(2n,\mathbb C)$)
unipotent representations attached to $\mathcal O$.
