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Representation Theory

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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
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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Additional Information

A. W. Knapp
Affiliation: Department of Mathematics, State University of New York, Stony Brook, New York 11794
MR Author ID: 103200

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