Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



The moment map for a multiplicity free action

Authors: Chal Benson, Joe Jenkins, Ronald L. Lipsman and Gail Ratcliff
Journal: Bull. Amer. Math. Soc. 31 (1994), 185-190
MSC: Primary 22C05; Secondary 22E30
MathSciNet review: 1260517
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let K be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space V. One calls this a multiplicity-free action whenever the K-isotypic components of $ \mathbb{C}{\text{[}}V]$ are K-irreducible. We have shown that this is the case if and only if the moment map $ \tau :V \to {\mathfrak{k}^{\ast} }$ for the action is finite-to-one on K-orbits. This is equivalent to a result concerning Gelfand pairs associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere.

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

  • [1] C. Benson, J. Jenkins, R. Lipsman, and G. Ratcliff. A geometric criterion for Gelfand pairs associated with the Heisenberg group, preprint. MR 1447402 (98i:22012)
  • [2] C. Benson, J. Jenkins, and G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), 85-116. MR 1000329 (90m:22022)
  • [3] G. Carcanno, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. 7 (1987), 1091-1105. MR 923441 (89h:22011)
  • [4] L. Corwin and F. Greenleaf, Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups, Pacific J. Math. 135 (1988), 233-267. MR 968611 (90b:22011a)
  • [5] I. M. Gelfand, Spherical functions on symmetric spaces, Dokl. Akad. Nauk USSR 70 (1950), 5-8; Amer. Math. Soc. Transl. Ser. 2, vol. 37, Amer. Math. Soc., Providence, RI, 1964, pp. 39-44. MR 0033832 (11:498b)
  • [6] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538. MR 664118 (83m:58040)
  • [7] -, Multiplicity free spaces, J. Differential Geom. 19 (1984), 31-56. MR 739781 (85h:58071)
  • [8] G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), 333-356. MR 665160 (84d:22019)
  • [9] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann. 290 (1991), 565-619. MR 1116239 (92j:17004)
  • [10] V. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190-213. MR 575790 (81i:17005)
  • [11] F. Knop, A Harish-Chandra homomorphism for reductive group actions, preprint. MR 1298713 (95h:14045)
  • [12] R. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pure Appl. 59 (1980), 337-374. MR 604474 (82b:22026)
  • [13] -, Orbital parameters for induced and restricted representations, Trans. Amer. Math. Soc. 313 (1989), 433-473. MR 930083 (90a:22008)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 22C05, 22E30

Retrieve articles in all journals with MSC: 22C05, 22E30

Additional Information

Keywords: Gelfand pairs, Heisenberg group, Orbit Method, moment map
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society