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Exceptional Unitary Representations
Of Semisimple Lie Groups


Author: A. W. Knapp
Journal: Represent. Theory 1 (1997), 1-24
MSC (1991): Primary 22E46, 22E47
DOI: https://doi.org/10.1090/S1088-4165-97-00001-0
Published electronically: November 4, 1996
MathSciNet review: 1429371
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Abstract: Let $G$ be a noncompact simple Lie group with finite center, let $K$ be a maximal compact subgroup, and suppose that $\text {rank }G=\text {rank }K$. If $G/K$ is not Hermitian symmetric, then a theorem of Borel and de Siebenthal gives the existence of a system of positive roots relative to a compact Cartan subalgebra so that there is just one noncompact simple root and it occurs exactly twice in the largest root. Let $\mathfrak {q}=\mathfrak {l}\oplus \mathfrak {u}$ be the $\theta $ stable parabolic obtained by building $\mathfrak {l}$ from the roots generated by the compact simple roots and by building $\mathfrak {u}$ from the other positive roots, and let $L\subseteq K$ be the normalizer of $\mathfrak {q}$ in $G$. Cohomological induction of an irreducible representation of $L$ produces a discrete series representation of $G$ under a dominance condition. This paper studies the results of this cohomological induction when the dominance condition fails. When the inducing representation is one-dimensional, a great deal is known about when the cohomologically induced representation is infinitesimally unitary. This paper addresses the question of finding Langlands parameters for the natural irreducible constituent of these representations, and also it finds some cases when the inducing representation is higher-dimensional and the cohomologically induced representation is infinitesimally unitary.


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  • [Ba] D. Barbasch, Representations with maximal primitive ideal, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (A. Connes, M. Duflo, A. Joseph, and R. Rentschler, eds.), Birkhäuser, Boston, 1990, pp. 317-331. MR 92j:17006
  • [Bi-Z] B. Binegar and R. Zierau, Unitarization of a singular representation of $SO(p,q)$, Commun. Math. Phys. 138 (1991), 245-258. MR 92h:22027
  • [Bo-deS] A. Borel and J. de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helvetici 23 (1949), 200-221. MR 11:326d
  • [Br-Ko] R. Brylinski and B. Kostant, Minimal representations, geometric quantization, and unitarity, Proc. Nat. Acad. Sci. USA 91 (1994), 6026-6029. MR 95d:58059
  • [EHW] T. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, Representation Theory of Reductive Groups (P. C. Trombi, ed.), Birkhäuser, Boston, 1983, pp. 97-143. MR 86c:22028
  • [EPWW] T. J. Enright, R. Parthasarathy, N. R. Wallach, and J. A. Wolf, Unitary derived functor modules with small spectrum, Acta Math. 154 (1985), 105-136. MR 86j:22026
  • [G-W] B. H. Gross and N. R. Wallach, A distinguished family of unitary representations for the exceptional groups of real $\text {rank}=4$, Lie Theory and Geometry: in Honor of Bertram Kostant (J.-L. Brylinski, R. Brylinski, V. Guillemin, and V. Kac, eds.), Birkhäuser, Boston, 1994, pp. 289-304. MR 96i:22034
  • [HC1] Harish-Chandra, Representations of semisimple Lie groups IV, Amer. J. Math. 77 (1955), 743-777; V, 78 (1956), 1-41; VI, 78 (1956), 564-628.
  • [HC2] Harish-Chandra, Harmonic analysis on real reductive groups I, J. Func. Anal. 19 (1975), 104-204. MR 53:3201
  • [Ja] H. P. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Func. Anal. 52 (1983), 385-412. MR 85a:17004
  • [Ka-S] D. Kazhdan and G. Savin, The smallest representations of simply laced groups, Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion on His Sixtieth Birthday, Part I (S. Gelbart, R. Howe, and P. Sarnak, eds.), Israel Math. Conf. Proc. 2, Weizmann Science Press of Israel, Jerusalem, 1990, pp. 209-223. MR 93f:22019
  • [Kn1] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, Princeton, N.J., 1986. MR 87j:22022
  • [Kn2] A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, Boston, 1996.
  • [Kn-Vo] A. W. Knapp and D. A. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, N.J., 1995. MR 96c:22023
  • [McG1] W. M. McGovern, Dixmier algebras and the orbit method, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (A. Connes, M. Duflo, A. Joseph, and R. Rentschler, eds.), Birkhäuser, Boston, 1990, pp. 397-416. MR 92f:17010
  • [McG2] W. M. McGovern, Rings of regular functions on nilpotent orbits II, Commun. in Algebra 22 (1994), 765-772. MR 95b:22035
  • [Vo1] D. A. Vogan, The algebraic structure of the representation of semisimple Lie groups I, Ann. of Math. 109 (1979), 1-60. MR 81j:22020
  • [Vo2] D. A. Vogan, Unitarizability of certain series of representations, Ann. of Math. 120 (1984), 141-187. MR 86h:22028
  • [W1] N. Wallach, The analytic continuation of the discrete series II, Trans. Amer. Math. Soc. 251 (1979), 19-37. MR 81a:22009
  • [W2] N. Wallach, On the unitarizability of derived functor modules, Invent. Math. 78 (1984), 131-141. MR 86f:22018

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Additional Information

A. W. Knapp
Affiliation: Department of Mathematics, State University of New York, Stony Brook, New York 11794
Email: aknapp@ccmail.sunysb.edu

DOI: https://doi.org/10.1090/S1088-4165-97-00001-0
Received by editor(s): June 19, 1996
Received by editor(s) in revised form: August 5, 1996
Published electronically: November 4, 1996
Additional Notes: Presented to the Society August 7, 1995 at the AMS Summer Meeting in Burlington, Vermont.
Article copyright: © Copyright 1997 American Mathematical Society

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