Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Irreducible Genuine Characters of the Metaplectic Group: Kazhdan-Lusztig Algorithm and Vogan Duality

Authors: David A. Renard and Peter E. Trapa
Journal: Represent. Theory 4 (2000), 245-295
MSC (2000): Primary 22E47; Secondary 22E50
Published electronically: July 31, 2000
MathSciNet review: 1795754
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We establish a Kazhdan-Lusztig algorithm to compute characters of irreducible genuine representations of the (nonlinear) metaplectic group with half-integral infinitesimal character. We then prove a character multiplicity duality theorem for representations of $Mp(2n,\mathbb R)$ at fixed half-integral infinitesimal character. This allows us to extend some of Langlands' ideas to $Mp(2n,\mathbb R)$.

References [Enhancements On Off] (What's this?)

  • [A] Adams, J., Lifting of Characters on Orthogonal and Metaplectic Groups, Duke Math. J., 92(1998), no. 1, 129-178. MR 99h:22014
  • [AB] Adams, J. and Barbasch, D., Genuine Representations of the Metaplectic Group, Compositio Math., 113(1998), no. 1, 23-60. MR 99h:22013
  • [ABV] Adams, J., D. Barbasch, and D. A. Vogan, Jr., The Langlands Classification and Irreducible Characters for Real Reductive Groups, Progress in Math, Birkhäuser (Boston), 104(1992). MR 93j:22001
  • [BBD] Beilinson, A., J. Bernstein and P. Deligne, Faisceaux Pervers, Astérisque, 100(1982), 5-171. MR 86g:32015
  • [GKM] Goresky, M., R. Kottwitz, and R. MacPherson, Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J., 89(1997), no. 3, 477-554. MR 99e:11064a; MR 99e:11064b
  • [H] Herb, R., Discrete series characters and two-structures, Trans. Amer. Math. Soc., 350(1998), no. 8, 3341-3369. MR 98k:22058
  • [La] Langlands, R.P., On the classification of irreducible representations of real algebraic groups, in Representation theory and harmonic analysis on semisimple Lie groups, Math. Survey Monographs, Amer. Math. Soc. (Providence), 33(1989), 101-170. MR 91e:22017
  • [LV] Lusztig, G. and Vogan, D., Singularities of closures of $K$-orbits on flag manifolds, Invent. Math. 71(1983), 365-379. MR 84h:14060
  • [MS] Mars, J.G.M. and Springer, T. A., Hecke Algebra Representations related to spherical varieties, Representation Theory, 2(1998), 33-69. MR 99h:20072
  • [MaO] Matsuki, T. and Oshima, T. Embeddings of discrete series into principal series, in The orbit method in representation theory (Copenhagen, 1988), Progress in Math., Birkhäuser(Boston), Boston, MA, 82(1990), 147-175. MR 92d:22020
  • [Mi1] Milicic, D., Intertwining functors and irreducibility of standard Harish-Chandra sheaves, in Harmonic analysis on reductive groups (Brunswick 1989), Progr. Math., Birkhäuser (Boston), 101(1991), 209-222. MR 94c:22013
  • [Mi2] Milicic, D., Notes on Localization Theory, preprint,
  • [R] Renard, D., Endoscopy for $Mp(2n,\mathbb{R} )$, Amer. J. Math. 121(1999), no. 6, 1215-1243. CMP 2000:05
  • [Tor] Torasso, P., Sur le Caractère de la Représentation de Shale-Weil de $Mp(n,\mathbb R)$ and $Sp(n,{\mathbb C})$, Math. Ann., 252(1980), 53-86. MR 82a:22019
  • [V1] Vogan, D. A., Jr., Irreducible characters of semisimple Lie groups I, Duke Math. J., 46(1979), no. 1, 61-108. MR 80g:22016
  • [V2] Vogan, D. A., Jr., Irreducible characters of semisimple Lie groups II: the Kazhdan-Lusztig conjectures, Duke Math. J., 46(1979), no. 4, 805-859. MR 81f:22024
  • [V3] Vogan, D. A., Jr., 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 84h:22036
  • [V4] Vogan, D. A., Jr., Irreducible characters of semisimple Lie groups IV: character multiplicity duality, Duke Math. J., 49(1982), no. 4, 943-1073. MR 84h:22037
  • [Vgr] Vogan, D. A., Jr., Representations of Real Reductive Lie Groups, Progress in Math., Birkhäuser(Boston), 15(1981). MR 83c:22022
  • [Ya] Yamamoto, A., Orbits in the flag variety and images of the moment map for classical groups I, Representation Theory, 1(1997), 329-494. MR 98j:22024

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 22E47, 22E50

Retrieve articles in all journals with MSC (2000): 22E47, 22E50

Additional Information

David A. Renard
Affiliation: University of Poitiers, Laboratoire de Mathématiques, BP 179, 86960 Futuroscope Cedex, France

Peter E. Trapa
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540
Address at time of publication: Department of Mathematics, Harvard University, Cambridge, MA 02138

Received by editor(s): November 12, 1999
Received by editor(s) in revised form: April 28, 2000
Published electronically: July 31, 2000
Additional Notes: The first author acknowledges the support of NSF grant DMS97-29992 and the Ellentuck Fund of the Institute for Advanced Study
The second author acknowledges the support of NSF grant DMS97-29995
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society