Small principal series and exceptional duality for two simply laced exceptional groups
Author:
Hadi Salmasian
Journal:
Trans. Amer. Math. Soc. 361 (2009), 19251947
MSC (2000):
Primary 22E46, 22E50, 11F27
Published electronically:
October 21, 2008
MathSciNet review:
2465824
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Abstract: We use the notion of rank defined in an earlier paper (2007) to introduce and study two correspondences between small irreducible unitary representations of the split real simple Lie groups of types , where , and two reductive classical groups. We show that these correspondences classify all of the unitary representations of rank two (in the sense of our earlier paper) of these exceptional groups. We study our correspondences for a specific family of degenerate principal series representations in detail.
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Yean Loke and Gordan
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R. Wallach, Transfer of unitary representations between real
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 [BSZ]
 Barchini, L.; Sepanski, M.; Zierau, R. Positivity of zeta distributions and small unitary representations. The ubiquitous heat kernel, 146, Contemp. Math., 398, Amer. Math. Soc., Providence, RI, 2006. MR 2218012 (2007d:22009)
 [Bo]
 Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 46. Translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin). SpringerVerlag, Berlin, 2002. xii+300 pp. MR 1890629 (2003a:17001)
 [DS]
 Dvorsky, Alexander; Sahi, Siddhartha Explicit Hilbert spaces for certain unipotent representations. II. Invent. Math. 138 (1999), no. 1, 203224. MR 1714342 (2001f:22041)
 [Ho]
 Howe, Roger On a notion of rank for unitary representations of the classical groups. Harmonic analysis and group representations, 223331, Liguori, Naples, 1982. MR 777342 (86j:22016)
 [HT]
 Howe, Roger E.; Tan, EngChye Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations. Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 1, 174. MR 1172839 (93j:22027)
 [GS]
 Gan, Wee Teck; Savin, Gordan On minimal representations definitions and properties. Represent. Theory 9 (2005), 4693 MR 2123125 (2006a:22015)
 [Ka]
 Kaneyuki, Soji The Sylvester's law of inertia in simple graded Lie algebras. J. Math. Soc. Japan 50 (1998), no. 3, 593614. MR 1626338 (99f:17035)
 [KS]
 Kazhdan, D.; Savin, G. The smallest representation of simply laced groups. Festschrift in honor of I. I. PiatetskiShapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), 209223, Israel Math. Conf. Proc., 2, Weizmann, Jerusalem, 1990. MR 1159103 (93f:22019)
 [Ki1]
 Kirillov, A. A. Unitary representations of nilpotent Lie groups. (Russian) Uspehi Mat. Nauk 17 (1962) no. 4 (106), 57110. MR 0142001 (25:5396)
 [Ki2]
 Kirillov, A. A. Lectures on the orbit method. Graduate Studies in Mathematics, 64. American Mathematical Society, Providence, RI, 2004. MR 2069175 (2005c:22001)
 [Kn1]
 Knapp, Anthony W. Lie groups beyond an introduction. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1399083 (98b:22002)
 [Kn2]
 Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examples. Reprint of the 1986 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 2001. MR 1880691 (2002k:22011)
 [Li1]
 Li, JianShu Singular unitary representations of classical groups. Invent. Math. 97 (1989), no. 2, 237255. MR 1001840 (90h:22021)
 [Li2]
 Li, JianShu On the classification of irreducible low rank unitary representations of classical groups. Compositio Math. 71 (1989), no. 1, 2948. MR 1008803 (90k:22027)
 [Mac]
 Mackey, George W. The theory of unitary group representations. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, Ill.London, 1976. MR 0396826 (53:686)
 [Sah]
 Sahi, Siddhartha Jordan algebras and degenerate principal series. J. Reine Angew. Math. 462 (1995), 118. MR 1329899 (96d:22022)
 [Sa1]
 Salmasian, H. A new notion of rank for unitary representations of reductive groups based on Kirillov's orbit method, Duke Math. J. 136 (2007), no. 1, 149. MR 2271294
 [Sa2]
 Salmasian, H. Isolatedness of the minimal representation and minimal decay of exceptional groups. Manuscripta Math. 120 (2006), no. 1, 3952. MR 2223480 (2007e:22010)
 [Sca]
 Scaramuzzi, R. A notion of rank for unitary representations of general linear groups. Trans. Amer. Math. Soc. 319 (1990), no. 1, 349379. MR 958900 (90i:22032)
 [SV]
 Springer, T. A.; Veldkamp, Ferdinand D. Octonions, Jordan algebras and exceptional groups. (English. English summary) Springer Monographs in Mathematics. SpringerVerlag, Berlin, 2000. MR 1763974 (2001f:17006)
 [LS]
 Loke, H. Y.; Savin, G. Rank and matrix coefficients for simply laced groups. J. Reine Angew. Math. 599 (2006), 201216. MR 2279102
 [To]
 Torasso, Pierre Méthode des orbites de KirillovDuflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle. (French) [The KirillovDuflo orbit method and minimal representations of simple groups over a local field of characteristic zero] Duke Math. J. 90 (1997), no. 2, 261377. MR 1484858 (99c:22028)
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 Wallach, Nolan R. Transfer of unitary representations between real forms. Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993), 181216, Contemp. Math., 177, Amer. Math. Soc., Providence, RI, 1994. MR 1303606 (95i:22024)
 [Ws]
 Weissman, Martin H. The FourierJacobi map and small representations. (English. English summary) Represent. Theory 7 (2003), 275299 (electronic). MR 1993361 (2004d:22014)
 [Zh]
 Zhang, Gen Kai Jordan algebras and generalized principal series representations. Math. Ann. 302 (1995), no. 4, 773786. MR 1343649 (97h:22012)
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Additional Information
Hadi Salmasian
Affiliation:
Department of Mathematics and Statistics, University of Windsor, Lambton Tower, 10th Floor, Windsor, Ontario, Canada N9B 3P4
Email:
hs79@uwindsor.ca
DOI:
http://dx.doi.org/10.1090/S0002994708045303
PII:
S 00029947(08)045303
Keywords:
Kirillov's orbit method,
Mackey analysis,
theta correspondence,
unitary representations
Received by editor(s):
October 10, 2006
Received by editor(s) in revised form:
March 14, 2007
Published electronically:
October 21, 2008
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
