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 $\tilde{K}=Spin(a,\mathbb{C})\times Spin(b,\mathbb{C})$ be the complexification of the maximal compact subgroup of $\widetilde{G_0}$. We consider the nilpotent orbit ${\mathcal{O}}_c$ parametrized by $[32^{2k}{1}^{2n-4k-3}]$ 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 $\tilde{K}$-spectra of the regular functions on certain real forms $\mathcal{O}$ of ${\mathcal{O}}_c$ transforming according to appropriate characters $\psi$ under $C_{\tilde{K}}(\mathcal{O})$, and then match them with the $\tilde{K}$-types of the genuine unipotent representations. The results provide instances for the orbit philosophy.