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Transactions of the American Mathematical Society

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

 

 

Finite- and infinite-dimensional representation of linear semisimple groups


Authors: James Lepowsky and Nolan R. Wallach
Journal: Trans. Amer. Math. Soc. 184 (1973), 223-246
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1973-0327978-7
MathSciNet review: 0327978
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Abstract: Every representation in the nonunitary principal series of a noncompact connected real semisimple linear Lie group G with maximal compact subgroup K is shown to have a K-finite cyclic vector. This is used to give a new proof of Harish-Chandra's theorem that every member of the nonunitary principal series has a (finite) composition series. The methods of proof are based on finite-dimensional G-modules, concerning which some new results are derived. Further related results on infinite-dimensional representations are also obtained.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0327978-7
Keywords: Finite-dimensional representation, infinite-dimensional representation, real semisimple linear Lie group, extensions of M-modules, nonunitary principal series, cyclic vectors, complete multiplicity, irreducible representation, composition series, universal enveloping algebra, subquotient theorem, Iwasawa decomposition
Article copyright: © Copyright 1973 American Mathematical Society