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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Exceptional Unitary Representations Of Semisimple Lie Groups
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by A. W. Knapp
Represent. Theory 1 (1997), 1-24
Published electronically: November 4, 1996


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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Bibliographic Information
  • A. W. Knapp
  • Affiliation: Department of Mathematics, State University of New York, Stony Brook, New York 11794
  • MR Author ID: 103200
  • Email:
  • 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.
  • © Copyright 1997 American Mathematical Society
  • Journal: Represent. Theory 1 (1997), 1-24
  • MSC (1991): Primary 22E46, 22E47
  • DOI:
  • MathSciNet review: 1429371