Embeddings of HarishChandra modules, homology and the composition series problem: the case of real rank one
Author:
David H. Collingwood
Journal:
Trans. Amer. Math. Soc. 285 (1984), 565579
MSC:
Primary 22E45; Secondary 20G05
MathSciNet review:
752491
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Abstract: Let be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup and form the (normalized) principal series representations . In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands' classification) all irreducible submodules and quotients of . As a corollary, all embeddings of an irreducible HarishChandra module into principal series are computed. The number of such embeddings is always less than or equal to three. These computations are equivalent to the determination of zero homology.
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 H. Hecht and W. Schmid, Characters, asymptotics and homology of HarishChandra modules, Acta Math. 151 (1983), 49151. MR 716371 (84k:22026)
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 , Composition factors of the principal series representations of the group , unpublished notes.
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 , Representations of real reductive Lie groups, Progress in Mathematics, Birkhäuser, Basel, 1981.
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 D. Zelobenko, Description of the quasisimple irreducible representations of the groups and , Math. USSRIzv. 11 (1977), 3150.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719840752491X
PII:
S 00029947(1984)0752491X
Keywords:
Representations of semisimple Lie groups,
embedding theorems,
composition series problem,
KazhdanLusztig conjectures
Article copyright:
© Copyright 1984
American Mathematical Society
