Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] 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)
  • [2] P. Cartier, et al., Séminaire ``Sophus Lie'' de L'École Normale Supérieure 1954/55, Théorie des algébres de Lie. Topologie des groups de Lie, Secrétariat mathématique, Paris, 1955. MR 17, 384.
  • [3] J. Dixmier, Idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple complexe, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1628-A1630. MR 0308225 (46:7339)
  • [4] Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185-243. MR 15, 100. MR 0056610 (15:100f)
  • [5] -, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26-65. MR 15, 398. MR 0058604 (15:398a)
  • [6] -, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485-528. MR 16, 111. MR 0063376 (16:111f)
  • [7] -, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119 (1965), 457-508. MR 31 #4862d. MR 0180631 (31:4862d)
  • [8] S. Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1-154. MR 41 #8587. MR 0263988 (41:8587)
  • [9] N. Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. MR 0143793 (26:1345)
  • [10] B. Kostant, On the existence and irreducibility of certain series of representations, Publication of 1971 Summer School in Math., edited by I. M. Gel'fand, Bolyai-Janós Math. Soc., Budapest (to appear). MR 0245725 (39:7031)
  • [11] J. Lepowsky, Multiplicity formulas for certain semisimple Lie groups, Bull. Amer. Math. Soc. 77 (1971), 600-605. MR 0301142 (46:300)
  • [12] -, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1-44. MR 0346093 (49:10819)
  • [13] H. Matsumoto, Quelques remarques sur les groupes de Lie algébriques réels, J. Math. Soc. Japan 16 (1964), 419-446. MR 32 #1292. MR 0183816 (32:1292)
  • [14] K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967), 383-429. MR 37 #1526. MR 0225936 (37:1526)
  • [15] C. Rader, Spherical functions on semisimple Lie groups, Ann. Sci. École Norm. Sup. (to appear).
  • [16] I. Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. (2) 71 (1960), 77-110. MR 22 #9546. MR 0118775 (22:9546)
  • [17] N. R. Wallach, Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107-113. MR 43 #7558. MR 0281844 (43:7558)
  • [18] -, Cyclic vectors and irreducibility for principal series representations. II, Trans. Amer. Math. Soc. 164 (1972), 389-396. MR 0320233 (47:8772)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E45

Retrieve articles in all journals with MSC: 22E45


Additional Information

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

American Mathematical Society