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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  .
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P. Želobenko, The analysis of irreducibility in the class of
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- H. Boerner, Darstellungen von Gruppen mit Berücksichtigung der Bedürfnisse der modernen Physik, Springer, Berlin, 1955; English transl., North-Holland, Amsterdam; Interscience, New York, 1963. MR 17, 710; MR 26 #6272. MR 0075211 (17:710b)
- [2]
- F. Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97-205. MR 18, 907. MR 0084713 (18:907i)
- [3]
- I. Gel'fand and M. Graev, Unitary representations of the real unimodular group (principal nondegenerate series), Izv. Akad. Nauk SSSR Ser. Mat. 17 (1953), 189-248; English transl., Amer. Math. Soc. Transl. (2) 2 (1956), 147-205. MR 15, 199; MR 17, 876. MR 0057261 (15:199a)
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- A. Knapp and E. Stein, Singular integrals and the principal series, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 281-284.
- [6]
- -, Singular integrals and the principal series. II, Proc. Nat. Acad. Sci. U.S.A. 65 (1970), 13-17.
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- B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642. MR 39 #7031. MR 0245725 (39:7031)
- [8]
- H. Matsumoto, Quelques remarques sur des groupes de Lie algébriques réels, J. Math. Soc. Japan 16 (1964), 419-446. MR 32 #1292. MR 0183816 (32:1292)
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- N. Wallach, Induced representations of Lie algebras and a theorem of Borel-Weil, Trans. Amer. Math. Soc. 136 (1969), 181-187. MR 38 #2258. MR 0233937 (38:2258)
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- D. P. Želobenko, The analysis of irreducibility in the class of elementary representations of a complex semisimple Lie group, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 108-133 = Math. USSR Izv. 2 (1968), 105-128. MR 37 #2906. MR 0227321 (37:2906)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1971-0281844-2
PII:
S 0002-9947(1971)0281844-2
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
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