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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Embeddings of Harish-Chandra modules, $ {\germ n}$-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 $ G$ be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup $ P = MAN$ and form the (normalized) principal series representations $ I_P^G(U)$. In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands' classification) all irreducible submodules and quotients of $ I_P^G(U)$. 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 $ {\mathfrak{n}}$-homology.

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Keywords: Representations of semisimple Lie groups, embedding theorems, composition series problem, Kazhdan-Lusztig conjectures
Article copyright: © Copyright 1984 American Mathematical Society

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