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)



Functional equations satisfied by intertwining operators of reductive groups

Author: Chen-bo Zhu
Journal: Trans. Amer. Math. Soc. 336 (1993), 881-899
MSC: Primary 22E46; Secondary 15A69, 22E30
MathSciNet review: 1097173
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper generalizes a recent work of Vogan and Wallach [VW] in which they derived a difference equation satisfied by intertwining operators of reductive groups. We show that, associated with each irreducible finite-dimensional representation, there is a functional equation relating intertwining operators. In this way, we obtain natural relations between intertwining operators for different series of induced representations.

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

  • [He] S. Helgason, Groups and geometric analysis, Academic Press, Orlando, Fla., 1984. MR 754767 (86c:22017)
  • [Hu] J. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math., vol. 9, Springer-Verlag, New York, 1987. MR 499562 (81b:17007)
  • [Kn] A. Knapp, Representation theory of semisimple Lie groups: An overview based on examples, Princeton Univ. Press, Princeton, N.J., 1986. MR 855239 (87j:22022)
  • [Ko] B. Kostant, On the tensor product of a finite and an infinite dimensional representation, J. Funct. Anal. 20 (1975), 257-285. MR 0414796 (54:2888)
  • [KS1] A. Knapp and E. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489-578. MR 0460543 (57:536)
  • [KS2] -, Intertwining operators for semisimple groups. II, Invent. Math. 60 (1980), 9-84. MR 582703 (82a:22018)
  • [Sl] S. Lang, $ S{l_2}(\mathbb{R})$, Graduate Texts in Math., vol. 105, Springer-Verlag, 1975.
  • [V1] D. Vogan, Jr., Representations of real reductive Lie groups, Birkhäuser, Boston, Mass., 1981. MR 632407 (83c:22022)
  • [VW] D. Vogan, Jr. and N. Wallach, Intertwining operators for real reductive groups, Adv. Math. 82 (1990), 203-243. MR 1063958 (91h:22022)
  • [W1] N. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker, New York, 1973. MR 0498996 (58:16978)
  • [Zh] C. Zhu, Two topics in harmonic analysis on reductive groups, Thesis, Yale University, 1990.
  • [Zu] G. Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math. (2) 106 (1977), 295-308. MR 0457636 (56:15841)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E46, 15A69, 22E30

Retrieve articles in all journals with MSC: 22E46, 15A69, 22E30

Additional Information

Keywords: Reductive groups, intertwining operators, functional equations
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society