Representations associated to small nilpotent orbits for real Spin groups

Abstract: 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.