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Representation Theory

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Strictly small representations and a reduction theorem for the unitary dual

Authors: Susana A. Salamanca-Riba and David A. Vogan Jr.
Journal: Represent. Theory 5 (2001), 93-110
MSC (2000): Primary 22E46
Published electronically: May 17, 2001
MathSciNet review: 1835000
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To any irreducible unitary representation $X$ of a real reductive Lie group we associate in a canonical way, a Levi subgroup $G_{su}$ and a representation of this subgroup. Assuming a conjecture of the authors on the infinitesimal character of $X$, we show that $X$ is cohomologically induced from a unitary representation of the subgroup $G_{su}$. This subgroup is in some cases smaller than the subgroup $G_{u}$ that the authors attached to $X$in earlier work. In those cases this provides a further reduction to the classification problem.

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

  • 1. J. Carmona, Sur la Classification des Modules Admissibles Irréductibles, in Non-commutative Harmonic Analysis and Lie Groups. (J. Carmona and M. Vergne, eds.), 11-34, Lecture Notes in Mathematics 1020, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. MR 85i:22022
  • 2. A. Knapp and D. A. Vogan Jr., Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, New Jersey, 1995. MR 96c:22033
  • 3. S. A. Salamanca-Riba and D. A. Vogan, Jr., On the Classification of Unitary Representations of Reductive Lie Groups, in Ann. of Math, 148 (1998), 1067-1133. MR 2000d:22017
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Additional Information

Susana A. Salamanca-Riba
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-0001

David A. Vogan Jr.
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): December 1, 2000
Received by editor(s) in revised form: March 30, 2001
Published electronically: May 17, 2001
Additional Notes: Supported in part by NSF grant DMS-9721441
Article copyright: © Copyright 2001 American Mathematical Society

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