Cyclic vectors and irreducibility for principal series representations.

Wallach, Nolan R.

doi:10.1090/S0002-9947-1971-0281844-2

Cyclic vectors and irreducibility for principal series representations.
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by Nolan R. Wallach

Trans. Amer. Math. Soc. 158 (1971), 107-113

DOI: https://doi.org/10.1090/S0002-9947-1971-0281844-2

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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 $SL(2n + 1,R),n \geqq 1$.

References

Hermann Boerner, Darstellungen von Gruppen mit Berücksichtigung der Bedürfnisse der modernen Physik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXIV, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (German). MR 0075211, DOI 10.1007/978-3-642-52808-8
François Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97–205 (French). MR 84713, DOI 10.24033/bsmf.1469
I. M. Gel′fand and M. I. Graev, Unitary representations of the real unimodular group (principal nondegenerate series), Izv. Akad. Nauk SSSR Ser. Mat. 17 (1953), 189–248 (Russian). MR 0057261
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455

Singular integrals and the principal series

63

Singular integrals and the principal series

65

Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
Hideya Matsumoto, Quelques remarques sur les groupes de Lie algébriques réels, J. Math. Soc. Japan 16 (1964), 419–446 (French). MR 183816, DOI 10.2969/jmsj/01640419
Nolan R. Wallach, Induced representations of Lie algebras and a theorem of Borel-Weil, Trans. Amer. Math. Soc. 136 (1969), 181–187. MR 233937, DOI 10.1090/S0002-9947-1969-0233937-4
D. P. Želobenko, The analysis of irreducibility in the class of elementary rupresentations of a semisimple complex Lie group, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 108–133 (Russian). MR 0227321