Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A spectral theoretic approach to the Kirillov-Duflo correspondence


Author: R. W. Raffoul
Journal: Proc. Amer. Math. Soc. 137 (2009), 2785-2794
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/S0002-9939-09-09916-X
Published electronically: April 6, 2009
MathSciNet review: 2497493
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Kirillov-Duflo orbit correspondance for compact Lie groups is parametrisation of the unitary dual, associating to the irreducible representation of highest weight $ \lambda$ the coadjoint orbit through $ \lambda +\delta$, where $ \delta$ is half the sum of the positive roots and justified by the character formulae of Weyl or Kirillov. In this paper we obtain this correspondence independently of character theory, showing that it arises from a convexity property of the Weyl functional calculus of the infinitesimal generators of the representation.


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

  • 1. E. ALBRECHT, Several variable spectral theory in the noncommutative case, Spectral theory (Warsaw, 1977), Banach Center Publ., 8, 9-30, PWN, Warsaw, 1982. MR 738273 (85h:47015)
  • 2. R.F.V. ANDERSON, The Weyl functional calculus, J. Functional Analysis, 4 (1969), 240-267. MR 0635128 (58:30405)
  • 3. D. ARNAL AND J. LUDWIG, La convexit$ \acute{e}$ de l'application moment d'un groupe de Lie, J. Functional Analysis, 105 (1992), 256-300. MR 1160080 (93j:22013)
  • 4. N. BERLINE, E. GETZLER AND M. VERGNE, Heat kernels and Dirac operators, Springer, Berlin, 2004. MR 2273508 (2007m:58033)
  • 5. F.F. BONSALL AND J. DUNCAN, Numerical ranges of operators on normed spaces and of elements of normed algebras, Cambridge University Press, London, 1971. MR 0288583 (44:5779)
  • 6. A.H. DOOLEY, R.W. RAFFOUL, Matrix coefficients and coadjoint orbits of compact Lie groups, Proc. Amer. Math. Soc., 135 (2007), 2567-2571. MR 2302577 (2008d:43006)
  • 7. D. FREED, M. HOPKINS AND C. TELEMAN, Loop groups and twisted $ K$-theory. II, preprint.
  • 8. L. HöRMANDER, The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. MR 1996773
  • 9. A.A. KIRILLOV, Characters of unitary representations of Lie groups, Funkcional. Anal. i Priložen. 2 (1968), 40-55. MR 0236318 (38:4615)
  • 10. K.-H. NEEB, Holomorphy and convexity in Lie theory, Walter de Gruyter & Co., Berlin, 2000. MR 1740617 (2001j:32020)
  • 11. E. NELSON, Operants: A functional calculus for non-commuting operators, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), Springer, New York, 1970, pp. 172-187. MR 0412857 (54:978)
  • 12. M.E. TAYLOR, Functions of several self-adjoint operators, Proc. Amer. Math. Soc., 19 (1968), 91-98. MR 0220082 (36:3149)
  • 13. M. VERGNE, Representations of Lie groups and the orbit method, Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), 59-101, Springer, New York, 1983. MR 713793 (85a:22022)
  • 14. N.J. WILDBERGER, The moment map of a Lie group representation, Trans. Amer. Math. Soc., 330 (1992), 257-268. MR 1040046 (92f:58064)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C40, 14E20, 46E25, 20C20

Retrieve articles in all journals with MSC (2000): 54C40, 14E20, 46E25, 20C20


Additional Information

R. W. Raffoul
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Email: raed@maths.unsw.edu.au

DOI: https://doi.org/10.1090/S0002-9939-09-09916-X
Keywords: Lie groups, coadjoint orbits
Received by editor(s): August 13, 2008
Published electronically: April 6, 2009
Communicated by: Varghese Mathai
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society