Embeddings of Harish-Chandra modules, -homology and the composition series problem: the case of real rank one

Author:
David H. Collingwood

Journal:
Trans. Amer. Math. Soc. **285** (1984), 565-579

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 Harish-Chandra 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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DOI:
https://doi.org/10.1090/S0002-9947-1984-0752491-X

Keywords:
Representations of semisimple Lie groups,
embedding theorems,
composition series problem,
Kazhdan-Lusztig conjectures

Article copyright:
© Copyright 1984
American Mathematical Society