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Tempered representations and nilpotent orbits


Author: Benjamin Harris
Journal: Represent. Theory 16 (2012), 610-619
MSC (2010): Primary 22E46; Secondary 43A65, 22E45
DOI: https://doi.org/10.1090/S1088-4165-2012-00414-9
Published electronically: December 13, 2012
MathSciNet review: 3001468
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a nilpotent orbit $ \mathcal {O}$ of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation $ \pi $ such that $ \mathcal {O}$ occurs in the wave front cycle of $ \pi $. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.


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  • 1. P. Bala, R. Carter, Classes of unipotent elements in simple algebraic groups. I, Math Proc. Cambridge. Philos. Soc., 79 (1976), 401-425. MR 0417306 (54:5363a)
  • 2. D. Barbasch, D. Vogan, The local structure of characters, J. Funct. Anal. 37 (1980), 27-55. MR 576644 (82e:22024)
  • 3. D. Collingwood, W. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, (1993), New York, NY. MR 1251060 (94j:17001)
  • 4. Harish-Chandra, Some results on an invariant integral on a semi-simple Lie algebra, Ann. of Math. (2), 80 (1964), 551-593. MR 0180629 (31:4862b)
  • 5. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, Volume 64, American Mathematical Society, (2004), Providence, RI. MR 2069175 (2005c:22001)
  • 6. C. Moeglin, Front d'Onde des representations des groupes classiques $ p$-adiques, American Journal of Mathematics, 118, no. 6 (1996), 1313-1346. MR 1420926 (98d:22015)
  • 7. C. Moeglin, J. L. Waldspurger, Modeles de Whittaker degeneres pour des groupes $ p$-adiques, Math Z. 196 (1987), 427-452. MR 913667 (89f:22024)
  • 8. A. Noël, Nilpotent orbits and theta stable parabolic subalgebras, Journal of Representation Theory, No. 2, (1998), 1-32. MR 1600330 (99g:17023)
  • 9. W. Rossmann, Limit orbits in reductive Lie algebras, Duke Math. J. 49 No. 1 (1982), 215-229. MR 650378 (84e:22021)
  • 10. W. Rossmann, Limit characters of reductive Lie groups, Invent. Math. 61 (1980), no. 1, 53-66. MR 587333 (81m:22021)
  • 11. W. Rossmann, Tempered representations and orbits, Duke Math. J. 49 No. 1 (1982), 231-247. MR 650379 (84m:22024)
  • 12. W. Rossmann, Picard-Lefschetz theory and characters of a semisimple Lie group, Invent. Math. 121 (1995), 579-611. MR 1353309 (96j:22017)
  • 13. W. Schmid, K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math. (2), 151 (2001), 1071-1118. MR 1779564 (2001j:22017)
  • 14. A. Yamamoto, Orbits in the flag variety and images of the moment map for classical groups I, Journal of Representation Theory, No. 1, (1997), 329-404. MR 1479152 (98j:22024)

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Additional Information

Benjamin Harris
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: blharris@math.mit.edu

DOI: https://doi.org/10.1090/S1088-4165-2012-00414-9
Keywords: Tempered representation, discrete series representation, wave front cycle, associated variety, reductive lie group, real reductive algebraic group, nilpotent orbit, distinguished nilpotent orbit, noticed nilpotent orbit, coadjoint orbit
Received by editor(s): October 19, 2010
Received by editor(s) in revised form: May 28, 2011, and September 18, 2011
Published electronically: December 13, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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