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Representation Theory

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Representations associated to small nilpotent orbits for complex Spin groups

Authors: Dan Barbasch and Wan-Yu Tsai
Journal: Represent. Theory 22 (2018), 202-222
MSC (2010): Primary 22E46, 22E47
Published electronically: October 25, 2018
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Abstract: This paper provides a comparison between the $ K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $ D$. Precisely, let $ G_0 =\operatorname {Spin}(2n,\mathbb{C})$ be the Spin complex group as a real group, and let $ K\cong G_0$ be the complexification of the maximal compact subgroup of $ G_0$. We compute $ K$-spectra of the regular functions on some small nilpotent orbits $ \mathcal {O}$ transforming according to characters $ \psi $ of $ C_{ K}(\mathcal {O})$ trivial on the connected component of the identity $ C_{ K}(\mathcal {O})^0$. We then match them with the $ {K}$-types of the genuine (i.e., representations which do not factor to $ \operatorname {SO}(2n,\mathbb{C})$) unipotent representations attached to $ \mathcal {O}$.

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Additional Information

Dan Barbasch
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14850

Wan-Yu Tsai
Affiliation: Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
Address at time of publication: Department of Mathematics and Statistics, University of Ottawa, Ontario, Canada

Keywords: Infinte dimensional representations, orthogonal groups, nilpotent orbits
Received by editor(s): September 5, 2017
Received by editor(s) in revised form: April 2, 2018
Published electronically: October 25, 2018
Additional Notes: The first author was supported in part by NSA Grant H98230-16-1-0006.
Article copyright: © Copyright 2018 American Mathematical Society

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