Finite- and infinite-dimensional representation of linear semisimple groups

Lepowsky, James; Wallach, Nolan R.

doi:10.1090/S0002-9947-1973-0327978-7

Finite- and infinite-dimensional representation of linear semisimple groups
HTML articles powered by AMS MathViewer

by James Lepowsky and Nolan R. Wallach PDF

Trans. Amer. Math. Soc. 184 (1973), 223-246 Request permission

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

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

Séminaire

Sophus Lie

de L’École Normale Supérieure

Théorie des algébres de Lie. Topologie des groups de Lie

17

Jacques 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 (French). MR 308225
Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185–243. MR 56610, DOI 10.1090/S0002-9947-1953-0056610-2
Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26–65. MR 58604, DOI 10.1090/S0002-9947-1954-0058604-0
Harish-Chandra, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485–528. MR 63376, DOI 10.1090/S0002-9947-1954-0063376-X
Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564. MR 180628, DOI 10.2307/2373023
Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
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
J. Lepowsky, Multiplicity formulas for certain semisimple Lie groups, Bull. Amer. Math. Soc. 77 (1971), 601–605. MR 301142, DOI 10.1090/S0002-9904-1971-12767-2
J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1–44. MR 346093, DOI 10.1090/S0002-9947-1973-0346093-X
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
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 225936, DOI 10.2307/1970351

Spherical functions on semisimple Lie groups

Ichirô Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. (2) 71 (1960), 77–110. MR 118775, DOI 10.2307/1969880
Nolan R. Wallach, Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107–113. MR 281844, DOI 10.1090/S0002-9947-1971-0281844-2
Nolan R. Wallach, Cyclic vectors and irreducibility for principal series representations. II, Trans. Amer. Math. Soc. 164 (1972), 389–396. MR 320233, DOI 10.1090/S0002-9947-1972-0320233-X