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Transactions of the American Mathematical Society

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



Cyclic vectors and irreducibility for principal series representations.

Author: Nolan R. Wallach
Journal: Trans. Amer. Math. Soc. 158 (1971), 107-113
MSC: Primary 22.60
MathSciNet review: 0281844
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Abstract: Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found. These cyclic vectors are used to obtain estimates for the number of irreducible subrepresentations of a principal series representations. The results are used to prove irreducibility for the full principal series of complex semisimple Lie groups and for $ SL(2n + 1,R),n \geqq 1$.

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Keywords: Semisimple Lie group, principal series representation, Hilbert space, invariant measure, unitary representation, cyclic vector, irreducible, Stone-Weierstrass theorem, Peter-Weyl theorem, Frobenius reciprocity, module, Weyl group, universal enveloping algebra
Article copyright: © Copyright 1971 American Mathematical Society