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)



On Gelfand pairs associated with solvable Lie groups

Authors: Chal Benson, Joe Jenkins and Gail Ratcliff
Journal: Trans. Amer. Math. Soc. 321 (1990), 85-116
MSC: Primary 22E25; Secondary 22D25, 43A20
MathSciNet review: 1000329
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a locally compact group, and let $ K$ be a compact subgroup of $ {\operatorname{Aut}}(G)$, the group of automorphisms of $ G$. There is a natural action of $ K$ on the convolution algebra $ {L^1}(G)$, and we denote by $ L_K^1(G)$ the subalgebra of those elements in $ {L^1}(G)$ that are invariant under this action. The pair $ (K,G)$ is called a Gelfand pair if $ L_K^1(G)$ is commutative. In this paper we consider the case where $ G$ is a connected, simply connected solvable Lie group and $ K \subseteq {\operatorname{Aut}}(G)$ is a compact, connected group. We characterize such Gelfand pairs $ (K,G)$, and determine a moduli space for the associated $ K$-spherical functions.

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

  • [BtD] T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1985. MR 781344 (86i:22023)
  • [Ca] G. Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Italiano 7 (1987), 1091-1105. MR 923441 (89h:22011)
  • [Di] H. Dib, Polynômes de Laguerre d'un argument matriciel, C. R. Acad. Sci. Paris 304 (1987), 111-114. MR 890627 (88j:33009)
  • [Ge] I. Gelfand, Spherical functions on symmetric spaces, Dokl. Akad. Nauk SSSR 70 (1950), 5-8. MR 0033832 (11:498b)
  • [He] S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
  • [Hz] C. Herz, Bessel functions of matrix argument, Ann. of Math 61 (1955), 474-523. MR 0069960 (16:1107e)
  • [Ho] R. Howe, Quantum mechanics and partial differential equations. J. Funct. Anal. 38 (1980), 188-255. MR 587908 (83b:35166)
  • [HR] A. Hulanicki and F. Ricci, A tauberian theorem and tangential convergence of bounded harmonic functions on balls in $ {{\mathbf{C}}^n}$, Invent. Math. 62 (1980), 325-331. MR 595591 (82e:32008)
  • [Je] J. Jenkins, Growth of connected locally compact groups, J. Funct. Anal. 12 (1973), 113-127. MR 0349895 (50:2388)
  • [Le] H. Leptin, A new kind of eigenfunction expansions on groups, Pacific J. Math. 116 (1985), 45-67. MR 769822 (86a:43011)
  • [Le2] -, On group algebras of nilpotent groups, Studia Math. 47 (1973), 37-49. MR 0330925 (48:9262)
  • [Lu] J. Ludwig, Polynomial growth and ideals in group algebras, Manuscripta Math. 30 (1980), 215-221. MR 557105 (81e:43012)
  • [Ka] V. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190-213. MR 575790 (81i:17005)
  • [KR] A. Kaplan and F. Ricci, Harmonic analysis on groups of Heisenberg type, Lecture Notes in Math., vol. 992, Springer, 1983, pp. 416-435. MR 729367 (85h:22017)
  • [Ki] A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Math. Surveys 17 (1978), 53-104. MR 0142001 (25:5396)
  • [Ma] G. Mackey, Unitary group representations in physics, Probability and Number Theory, Benjamin-Cummings, 1978. MR 515581 (80i:22001)
  • [Na] M. Naimark, Normed rings, Wolters-Noordhoff, 1970. MR 0355601 (50:8075)
  • [Ta] M. Taylor, Noncommutative harmonic analysis, Math. Surveys, no. 22, Amer. Math. Soc., Providence, R.I., 1986. MR 852988 (88a:22021)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E25, 22D25, 43A20

Retrieve articles in all journals with MSC: 22E25, 22D25, 43A20

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society