
Journal of Lie Theory 28 (2018), No. 4, 9871042 Copyright Heldermann Verlag 2018 Representations Associated to Small Nilpotent Orbits for Real Spin Groups Dan Barbasch Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A. dmb14@cornell.edu WanYu Tsai Institute of Mathematics, Academia Sinica 6F, Roosevelt Road, Taipei 10617, Taiwan wytsai@math.sinica.edu.tw [Abstractpdf] \newcommand{\tu}{\widetilde} \newcommand{\bbC}{{\mathbb{C}}} \newcommand{\calO}{{\mathcal{O}}} 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 $\tu{G_0} =\tu{Spin}(a,b)$ with $a+b=2n$, the nonlinear double cover of $Spin(a,b)$, and let $\tu{K}=Spin(a, \bbC)\times Spin(b, \bbC)$ be the complexification of the maximal compact subgroup of $\tu{G _0}$. We consider the nilpotent orbit $\calO_c$ parametrized by $[3 \ 2^{2k} \ 1^{2n4k3}]$ 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 $\tu{K}$spectra of the regular functions on certain real forms $\calO$ of $\calO_c$ transforming according to appropriate characters $\psi$ under $C_{\tu{K}}(\calO)$, and then match them with the $\tu{K}$types of the genuine unipotent representations. The results provide instances for the orbit philosophy. Keywords: Spin groups, nilpotent orbits, unipotent representations. MSC: 22E47 [ Fulltextpdf (348 KB)] for subscribers only. 