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Limits of discrete series with infinitesimal character zero

Authors: Henri Carayol and A. W. Knapp
Journal: Trans. Amer. Math. Soc. 359 (2007), 5611-5651
MSC (2000): Primary 20G20, 22E45; Secondary 14L35.
Published electronically: June 22, 2007
MathSciNet review: 2327045
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Abstract: For a connected linear semisimple Lie group $ G$, this paper considers those nonzero limits of discrete series representations having infinitesimal character 0, calling them totally degenerate. Such representations exist if and only if $ G$ has a compact Cartan subgroup, is quasisplit, and is acceptable in the sense of Harish-Chandra.

Totally degenerate limits of discrete series are natural objects of study in the theory of automorphic forms: in fact, those automorphic representations of adelic groups that have totally degenerate limits of discrete series as archimedean components correspond conjecturally to complex continuous representations of Galois groups of number fields. The automorphic representations in question have important arithmetic significance, but very little has been proved up to now toward establishing this part of the Langlands conjectures.

There is some hope of making progress in this area, and for that one needs to know in detail the representations of $ G$ under consideration. The aim of this paper is to determine the classification parameters of all totally degenerate limits of discrete series in the Knapp-Zuckerman classification of irreducible tempered representations, i.e., to express these representations as induced representations with nondegenerate data.

The paper uses a general argument, based on the finite abelian reducibility group $ R$ attached to a specific unitary principal series representation of $ G$. First an easy result gives the aggregate of the classification parameters. Then a harder result uses the easy result to match the classification parameters with the representations of $ G$ under consideration in representation-by-representation fashion. The paper includes tables of the classification parameters for all such groups $ G$.

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Additional Information

Henri Carayol
Affiliation: Université Louis Pasteur et C.N.R.S., I.R.M.A., 7 Rue René Descartes, 67084 Strasbourg Cedex, France

A. W. Knapp
Affiliation: Department of Mathematics, State University of New York, Stony Brook, New York 11794
Address at time of publication: 81 Upper Sheep Pasture Road, East Setauket, New York 11733–1729

Keywords: Semisimple Lie group, discrete series, tempered representations.
Received by editor(s): January 5, 2006
Published electronically: June 22, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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