|
Matrix coefficients and coadjoint orbits of compact Lie groups
Author(s):
A.
H.
Dooley;
R.
W.
Raffoul
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2567-2571.
MSC (2000):
Primary 43A77, 22E99;
Secondary 47Nxx
Posted:
March 22, 2007
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let  be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the  -function and the pull-back of an arbitrary unitary irreducible representation of  to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.
References:
-
- 1.
- R. F. V. Anderson, The Weyl functional calculus, J. Functional Analysis, 4 (1969), 240-267. MR 0635128 (58:30405)
- 2.
- D. Arnal and J. Ludwig, La convexit
 de l'application moment d'un groupe de Lie, J. Functional Analysis, 4 (1992), 256-300.MR 1160080 (93j:22013) - 3.
- V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin, 1980.MR 0997295 (90c:58046)
- 4.
- F. Cazzaniga, Kirillov's formula for noncentral functions on
 , Rend. Sem. Mat. Univ. Politec. Torino, 50 (1992), 233-242.MR 1249464 (94m:22012) - 5.
- A. H. Dooley and N. J. Wildberger, Global character formulae for compact Lie groups, Trans. Amer. Math. Soc., 351 (1999), 477-495.MR 1638234 (99i:22013)
- 6.
- V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 77 (1984), 533-546. MR 0759258 (86b:58042a)
- 7.
- Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math., 79 (1957), 87-120. MR 0084104 (18:809d)
- 8.
- B. Jefferies, The Weyl calculus for Hermitian matrices, Proc. Amer. Math. Soc., 124 (1996), 121-128. MR 1301032 (96d:47023)
- 9.
- M. S. Khalgui, Caractères des groupes de Lie, J. Funct. Anal., 47 (1982), 64-77.MR 0663833 (84f:22020)
- 10.
- A. A. Kirillov, Characters of unitary representations of Lie groups, Funkcional. Anal. i Prilozen, 2 (1968), 40-55. MR 0236318 (38:4615)
- 11.
- B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Annals sci.
 cole norm. sup. (4), 6 (1973), 413-455.MR 0364552 (51:806) - 12.
- J.-J. Loeb, Formule de Kirillov pour les groupes de Lie semi-simples compacts, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973-75), 230-256. Lecture Notes in Math., Vol. 497, Springer, Berlin, 1975.MR 0407204 (53:10987)
- 13.
- K.-H. Neeb, Holomorphy and convexity in Lie theory, Walter de Gruyter & Co., Berlin, 2000.MR 1740617 (2001j:32020)
- 14.
- 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)
- 15.
- L. Pukánszky, On the characters and the Plancherel formula of nilpotent groups, J. Functional Analysis, 1 (1976), 255-280.MR 0228656 (37:4236)
- 16.
- W. Rossmann, Kirillov's character formula for reductive Lie groups, Invent. Math., 48 (1978), 207-220. MR 0508985 (81g:22012)
- 17.
- R. Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math., 138 (1998), 46-91. MR 1645052 (2000a:53148)
- 18.
- 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):
43A77, 22E99,
47Nxx
Retrieve articles in all Journals with MSC
(2000):
43A77, 22E99,
47Nxx
Additional Information:
A.
H.
Dooley
Affiliation:
School of Mathematics, University of New South Wales, Sydney NSW 2000, Australia
Email:
a.dooley@unsw.edu.au
R.
W.
Raffoul
Affiliation:
School of Mathematics, University of New South Wales, Sydney NSW 2000, Australia
Email:
raed@maths.unsw.edu.au
DOI:
10.1090/S0002-9939-07-08781-3
PII:
S 0002-9939(07)08781-3
Keywords:
Coadjoint orbits,
Lie groups,
matrix coefficients,
moment map,
Weyl functional calculus.
Received by editor(s):
April 18, 2006
Posted:
March 22, 2007
Additional Notes:
The authors gratefully acknowledge the support of the Australian Research Council.
Communicated by:
Jane M. Hawkins
Copyright of article:
Copyright
2007,
American Mathematical Society
|
Comments: webmaster@ams.org