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

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Abstract: Let be a real semisimple Lie algebra with Iwasawa decomposition , and let be the centralizer of in . A conical vector in a -module is defined to be a nonzero -invariant vector. The -modules which are algebraically induced from one-dimensional -modules on which the action of is trivial have ``canonical generators'' which are conical vectors. In this paper, all the conical vectors in these -modules are found, in the special case . 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.

**[1]**I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand,*Differential operators on the fundamental affine space*, Dokl. Akad. Nauk SSSR**195**(1970), 1255–1258 (Russian). MR**0277669****[2]**Jacques Dixmier,*Algèbres enveloppantes*, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). Cahiers Scientifiques, Fasc. XXXVII. MR**0498737****[3]**Michel Duflo,*Représentations irréductibles des groupes semi-simples complexes*, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973–75), Springer, Berlin, 1975, pp. 26–88. Lecture Notes in Math., Vol. 497 (French). MR**0399353****[4]**Harish-Chandra,*Representations of semisimple Lie groups. II*, Trans. Amer. Math. Soc.**76**(1954), 26–65. MR**0058604**, https://doi.org/10.1090/S0002-9947-1954-0058604-0**[5]**SigurÄ‘ur Helgason,*A duality for symmetric spaces with applications to group representations*, Advances in Math.**5**(1970), 1–154 (1970). MR**0263988**, https://doi.org/10.1016/0001-8708(70)90037-X**[6]**Bertram Kostant,*On the existence and irreducibility of certain series of representations*, Bull. Amer. Math. Soc.**75**(1969), 627–642. MR**0245725**, https://doi.org/10.1090/S0002-9904-1969-12235-4**[7]**J. Lepowsky,*Algebraic results on representations of semisimple Lie groups*, Trans. Amer. Math. Soc.**176**(1973), 1–44. MR**0346093**, https://doi.org/10.1090/S0002-9947-1973-0346093-X**[8]**G. D. Mostow,*Strong rigidity of locally symmetric spaces*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. MR**0385004****[9]**C. Rader,*Spherical functions on semisimple Lie groups*, Thesis and unpublished supplements, University of Washington, 1971.**[10]**Daya-Nand Verma,*Structure of certain induced representations of complex semisimple Lie algebras*, Bull. Amer. Math. Soc.**74**(1968), 160–166. MR**0218417**, https://doi.org/10.1090/S0002-9904-1968-11921-4**[11]**Nolan R. Wallach,*Harmonic analysis on homogeneous spaces*, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 19. MR**0498996****[12]**M. Hu,*Determination of the conical distributions for rank one symmetric spaces*, Thesis, Massachusetts Institute of Technology, 1973.

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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