Journal of Lie Theory

no. 4
Representations Associated to Small Nilpotent Orbits for Real Spin Groups
Dan Barbasch1; Wan-Yu Tsai2
1Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.
2Institute of Mathematics, Academia Sinica 6F, Roosevelt Road, Taipei 10617, Taiwan
Journal of Lie Theory, Volume 28 (2018) no. 4, pp. 987-1042
\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.
DOI: 10.5802/jolt.1037
Classification: 22E47
Keywords: Spin groups, nilpotent orbits, unipotent representations

Dan Barbasch  1 ; Wan-Yu Tsai  2

1 Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.
2 Institute of Mathematics, Academia Sinica 6F, Roosevelt Road, Taipei 10617, Taiwan 
Dan Barbasch; Wan-Yu Tsai. Representations Associated to Small Nilpotent Orbits for Real Spin Groups. Journal of Lie Theory, Volume 28 (2018) no. 4, pp. 987-1042. doi: 10.5802/jolt.1037
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     title = {Representations {Associated} to {Small} {Nilpotent} {Orbits} for {Real} {Spin} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {987--1042},
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