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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Small unitary representations of the double cover of $ \operatorname{SL}(m)$


Author: Adam R. Lucas
Journal: Trans. Amer. Math. Soc. 360 (2008), 3153-3192
MSC (2000): Primary 22E46; Secondary 22E15
Published electronically: January 29, 2008
MathSciNet review: 2379792
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Abstract: The irreducible unitary representations of the double cover $ \widetilde{\mathrm{SL}(m)}$ of the real group $ \mathrm{SL}(m)$, with infinitesimal character $ \frac{1}{2}\rho$, which are small in the sense that their annihilator in the universal enveloping algebra is maximal, are expressed as Langlands quotients of generalized principal series. In the case where $ m$ is even we show that there are four such representations and in the case where $ m$ is odd there is just one. The representations' smallness allows them to be written as a sum of virtual representations, leading to a character formula for their $ K$-types. We investigate the place of these small representations in the orbit method and, in the case of $ \widetilde{\mathrm{SL}(2l+1)}$, show that the representation is attached to a nilpotent coadjoint orbit.The $ K$-type spectrum for the Langlands quotients is explicitly determined and shown to be multiplicity free.


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Additional Information

Adam R. Lucas
Affiliation: Department of Mathematics, Saint Mary’s College of California, P.O. Box 3517, Moraga, California 94575-3517
Email: arl3@stmarys-ca.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04401-2
PII: S 0002-9947(08)04401-2
Received by editor(s): April 11, 2005
Received by editor(s) in revised form: May 27, 2006
Published electronically: January 29, 2008
Additional Notes: The author is thankful to his advisor, Professor David Vogan, for his guidance and endless patience, as well as Peter Trapa and Thom Pietraho for helpful discussions
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.