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)

 
 

 

Conical vectors in induced modules


Author: J. Lepowsky
Journal: Trans. Amer. Math. Soc. 208 (1975), 219-272
MSC: Primary 17B15
DOI: https://doi.org/10.1090/S0002-9947-1975-0376786-1
MathSciNet review: 0376786
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$, and let $ \mathfrak{m}$ be the centralizer of $ \mathfrak{a}$ in $ \mathfrak{k}$. A conical vector in a $ \mathfrak{g}$-module is defined to be a nonzero $ \mathfrak{m} \oplus \mathfrak{n}$-invariant vector. The $ \mathfrak{g}$-modules which are algebraically induced from one-dimensional $ (\mathfrak{m} \oplus \mathfrak{a} \oplus \mathfrak{n})$-modules on which the action of $ \mathfrak{m}$ is trivial have ``canonical generators'' which are conical vectors. In this paper, all the conical vectors in these $ \mathfrak{g}$-modules are found, in the special case $ \dim \mathfrak{a} = 1$. The conical vectors have interesting expressions as polynomials in two variables which factor into linear or quadratic factors. Because it is too difficult to determine the conical vectors by direct computation, metamathematical ``transfer principles'' are proved, to transfer theorems about conical vectors from one Lie algebra to another; this reduces the problem to a special case which can be solved. The whole study is carried out for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero. An exposition of the Kostant-Mostow double transitivity theorem is included.


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

  • [1] I. N. Bernšteĭn, I. M. Gel'fand and S. I. Gel'fand, (a) Structure of representations generated by highest weight vectors, Funkcional. Anal. i Priložen. 5 (1971), 1-9 = Functional Anal. Appl. 5 (1971), 1-8. (b) Differential operators on the fundamental affine space, Dokl. Akad. Nauk SSSR 195 (1970), 1255-1258. (Russian) MR 43 #3402. MR 0277669 (43:3402)
  • [2] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. MR 0498737 (58:16803a)
  • [3] M. Duflo, Représentations irréductibles des groupes semi-simples complexes (to appear). MR 0399353 (53:3198)
  • [4] Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26-65. MR 15, 398. MR 0058604 (15:398a)
  • [5] S. Helgason, (a) A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1-154. MR 41 #8587. (b) Analysis on Lie groups and homogeneous spaces, CBMS Regional Conference Series in Math., no. 14, Amer. Math. Soc., Providence, R. I., 1972. MR 47 #5179. MR 0263988 (41:8587)
  • [6] B. Kostant, On the existence and irreducibility of certain series of representations, Publ. 1971 Summer School in Math., edited by I. M. Gel'fand, Bolyai-Janós Math. Soc., Budapest (to appear). MR 0245725 (39:7031)
  • [7] J. Lepowsky, (a) Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc 176 (1973), 1-44. (b) On the Harish-Chandra homomorphism, Trans. Amer. Math. Soc. 208 (1975), 193-218. (c) Uniqueness of embeddings of certain induced modules (to appear). (d) On the uniqueness of conical vectors (to appear). (e) A generalization of H. Weyl's ``unitary trick", Trans. Amer. Math. Soc. (to appear). MR 0346093 (49:10819)
  • [8] G. D. Mostow, Rigidity of locally symmetric spaces, Ann. of Math. Studies, no. 78, Princeton Univ. Press, Princeton, N. J., 1973. MR 0385004 (52:5874)
  • [9] C. Rader, Spherical functions on semisimple Lie groups, Thesis and unpublished supplements, University of Washington, 1971.
  • [10] D.-N. Verma, (a) Structure of certain induced representations of complex semisimple Lie algebras, Thesis, Yale University, 1966. (b) Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968), 160-166; errata, p. 628. MR 36 #1503; #5182. MR 0218417 (36:1503)
  • [11] N. R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Appl. Math., vol. 19, Dekker, New York, 1973. MR 0498996 (58:16978)
  • [12] M. Hu, Determination of the conical distributions for rank one symmetric spaces, Thesis, Massachusetts Institute of Technology, 1973.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B15

Retrieve articles in all journals with MSC: 17B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0376786-1
Keywords: Conical vectors, highest weight vectors, induced modules, Verma modules, real semisimple Lie algebras, real rank one, semisimple symmetric Lie algebras, splitting Cartan subspaces, restricted roots, restricted weight vectors, restricted Weyl group, universal enveloping algebra, double transitivity theorem, polynomial invariants
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society