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Vogan duality for nonlinear type B


Author: Scott Crofts
Journal: Represent. Theory 15 (2011), 258-306
MSC (2010): Primary 20G05
DOI: https://doi.org/10.1090/S1088-4165-2011-00398-8
Published electronically: March 24, 2011
MathSciNet review: 2788895
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Abstract: Let $ \mathbb{G}=\mathrm{Spin}[4n+1]$ be the connected, simply connected complex Lie group of type $ B_{2n}$ and let $ G=\mathrm{Spin}(p,q)$ $ (p+q=4n+1)$ denote a (connected) real form. If $ q \notin \left\{0,1\right\}$, $ G$ has a nontrivial fundamental group and we denote the corresponding nonalgebraic double cover by $ \tilde{G}=\widetilde{\mathrm{Spin}}(p,q)$. The main purpose of this paper is to describe a symmetry in the set of genuine parameters for the various $ \tilde{G}$ at certain half-integral infinitesimal characters. This symmetry is used to establish a duality of the corresponding generalized Hecke modules and ultimately results in a character multiplicity duality for the genuine characters of $ \tilde{G}$.


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  • 1. J. Adams, D. Barbasch, A. Paul, P. Trapa, and D. A. Vogan, Jr., Unitary Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007), no. 3, 701-751 (electronic). MR 2291917
  • 2. Jeffrey Adams and Fokko du Cloux, Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu 8 (2009), no. 2, 209-259. MR 2485793
  • 3. Jeffrey Adams and Rebecca. Herb, Lifting of characters for nonlinear simply laced groups, Representation Theory 14 (2010), 70-147 (electronic).
  • 4. Jeffrey Adams and Peter E. Trapa, Duality for nonlinear simply laced groups, Preprint, March 2007.
  • 5. Alexandre Beĭlinson and Joseph Bernstein, Localisation de $ g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15-18. MR 610137 (82k:14015)
  • 6. J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387-410. MR 632980 (83e:22020)
  • 7. Bill Casselman, Computations in real tori, Representation Theory of Real Reductive Lie Groups (Snowbird, Utah, 2006), Contemporary Mathematics, vol. 472, Amer. Math. Soc., Providence, RI, 2008, pp. 137-152. MR 2454333 (2010g:22013)
  • 8. David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412 (81j:20066)
  • 9. Anthony W. Knapp, Lie groups beyond an introduction, second ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389 (2003c:22001)
  • 10. David A. Renard and Peter E. Trapa, Irreducible genuine characters of the metaplectic group: Kazhdan-Lusztig algorithm and Vogan duality, Represent. Theory 4 (2000), 245-295 (electronic). MR 1795754 (2001m:22031)
  • 11. -, Kazhdan-Lusztig algorithms for nonlinear groups and applications to Kazhdan-Patterson lifting, Amer. J. Math. 127 (2005), no. 5, 911-971. MR 2170136 (2006g:22011)
  • 12. David A. Vogan, Jr., Irreducible characters of semisimple Lie groups. I, Duke Math. J. 46 (1979), no. 1, 61-108. MR 523602 (80g:22016)
  • 13. -, Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures, Duke Math. J. 46 (1979), no. 4, 805-859. MR 552528 (81f:22024)
  • 14. -, Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser Boston, Mass., 1981. MR 632407 (83c:22022)
  • 15. -, Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J. 49 (1982), no. 4, 943-1073. MR 683010 (84h:22037)
  • 16. -, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math. 71 (1983), no. 2, 381-417. MR 689650 (84h:22036)
  • 17. -, The Kazhdan-Lusztig conjecture for real reductive groups, Representation theory of reductive groups (Park City, Utah, 1982), Progress in Mathematics, vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 223-264. MR 733817 (85g:22028)

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Additional Information

Scott Crofts
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064

DOI: https://doi.org/10.1090/S1088-4165-2011-00398-8
Received by editor(s): August 13, 2009
Received by editor(s) in revised form: August 5, 2010
Published electronically: March 24, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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