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Journal of Lie Theory 28 (2018), No. 4, 987--1042
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.

Wan-Yu Tsai
Institute of Mathematics, Academia Sinica 6F, Roosevelt Road, Taipei 10617, Taiwan


\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 $[3 \ 2^{2k} \ 1^{2n-4k-3}]$ 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 $\tu{K}$-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

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