Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T11:43:28.532Z Has data issue: false hasContentIssue false
  • English
  • Français

Duality for nonlinear simply laced groups

Part of: Lie groups

Published online by Cambridge University Press:  19 March 2012

Jeffrey Adams  and
Peter E. Trapa
Jeffrey Adams
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (email: jda@math.umd.edu)
Peter E. Trapa
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA (email: ptrapa@math.utah.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish a character multiplicity duality for a certain natural class of nonlinear (nonalgebraic) groups arising as two-fold covers of simply laced real reductive algebraic groups. This allows us to extend part of the formalism of the local Langlands conjecture to such groups.

Keywords

local Langlands conjecturenonlinear groupsKazhdan–Lusztig theory

MSC classification

Secondary: 22E47: Representations of Lie and real algebraic groups
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 931 - 965
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Ada91]Adams, J., Lifting of characters, Progress in Mathematics, vol. 101 (Birkhäuser, Boston, MA, 1991).CrossRefGoogle Scholar
[Ada98]Adams, J., Lifting of characters on orthogonal and metaplectic groups, Duke Math. J. 92 (1998), 129–178.CrossRefGoogle Scholar
[Ada04]Adams, J., Nonlinear covers of real groups, Int. Math. Res. Not. IMRN 2004 (2004), 4031–4047.CrossRefGoogle Scholar
[ABPTV07]Adams, J., Barbasch, D., Paul, A., Trapa, P. E. and Vogan, D. A. Jr, Unitary Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007), 701–751.CrossRefGoogle Scholar
[ABV92]Adams, J., Barbasch, D. and Vogan, D. A. Jr, The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104 (Birkhäuser, Boston, MA, 1992).CrossRefGoogle Scholar
[AH10]Adams, J. and Herb, R. A., Lifting of characters for non-linear real groups, Represent. Theory 14 (2010), 70–147.CrossRefGoogle Scholar
[Cro11]Crofts, S., Vogan duality for nonlinear type B, Represent. Theory 15 (2011), 258–306.CrossRefGoogle Scholar
[Kna86]Knapp, A. W., Representation theory of semisimple groups. An overview based on examples (Princeton University Press, Princeton, NJ, 1986).CrossRefGoogle Scholar
[KV95]Knapp, A. W. and Vogan, D. A. Jr, Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45 (Princeton University Press, Princeton, NJ, 1995).CrossRefGoogle Scholar
[Lan89]Langlands, R. P., On the classification of irreducible representations of real algebraic groups, in Representation theory and harmonic analysis on semisimple Lie groups, Mathematical Surveys Monographs, vol. 31 (American Mathematical Society, Providence, RI, 1989), 101–170.CrossRefGoogle Scholar
[Mir86]Mirkovic, I., Classification of irreducible tempered representations, PhD thesis, University of Utah (1986).Google Scholar
[PR94]Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, 1994), translated by Rachel Rowen from the 1991 Russian original.Google Scholar
[RT00]Renard, D. A. and Trapa, P. E., Irreducible genuine characters of the metaplectic group: Kazhdan–Lusztig algorithm and Vogan duality, Represent. Theory 4 (2000), 245–295.CrossRefGoogle Scholar
[RT03]Renard, D. A. and Trapa, P. E., Irreducible characters of the metaplectic group. II. Functoriality, J. Reine Angew. Math. 557 (2003), 121–158.Google Scholar
[RT05]Renard, D. A. and Trapa, P. E., Kazhdan–Lusztig algorithms for nonlinear groups and applications to Kazhdan–Patterson lifting, Amer. J. Math. 127 (2005), 911–971.CrossRefGoogle Scholar
[Ste62]Steinberg, R., Générateurs, relations et revêtements de groupes algébriques, in Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) (Librairie Universitaire, Louvain, 1962), 113–127.Google Scholar
[Vog79]Vogan, D. A. Jr, Irreducible characters of semisimple Lie groups. I, Duke Math. J. 46 (1979), 61–108.Google Scholar
[Vog81]Vogan, D. A. Jr, Representations of real reductive Lie groups, Progress in Mathematics, vol. 15 (Birkhäuser, Boston, MA, 1981).Google Scholar
[Vog82]Vogan, D. A. Jr, Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J. 49 (1982), 943–1073.CrossRefGoogle Scholar
[Vog83]Vogan, D. A. Jr, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan–Lusztig conjecture in the integral case, Invent. Math. 71 (1983), 381–417.CrossRefGoogle Scholar
[Vog84]Vogan, D. A. Jr, Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), 141–187.CrossRefGoogle Scholar
[Vog93]Vogan, D. A. Jr, The local Langlands conjecture, in Representation theory of groups and algebras, Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), 305–379.CrossRefGoogle Scholar