Matrix coefficients and coadjoint orbits of compact Lie groups

Dooley, A.; Raffoul, R.

doi:10.1090/S0002-9939-07-08781-3

Matrix coefficients and coadjoint orbits of compact Lie groups

Authors: A. H. Dooley and R. W. Raffoul
Journal: Proc. Amer. Math. Soc. 135 (2007), 2567-2571
MSC (2000): Primary 43A77, 22E99; Secondary 47Nxx
DOI: https://doi.org/10.1090/S0002-9939-07-08781-3
Published electronically: March 22, 2007
MathSciNet review: 2302577
Full-text PDF

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.

1. Robert F. V. Anderson, The Weyl functional calculus, J. Functional Analysis 4 (1969), 240–267. MR 0635128
2. D. Arnal and J. Ludwig, La convexité de l’application moment d’un groupe de Lie, J. Funct. Anal. 105 (1992), no. 2, 256–300 (French, with English summary). MR 1160080, https://doi.org/10.1016/0022-1236(92)90080-3
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 𝑆𝑈(2), Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), no. 3, 233–242 (1993). MR 1249464
5. A. H. Dooley and N. J. Wildberger, Global character formulae for compact Lie groups, Trans. Amer. Math. Soc. 351 (1999), no. 2, 477–495. MR 1638234, https://doi.org/10.1090/S0002-9947-99-02406-X
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, https://doi.org/10.2307/2372387
8. Brian Jefferies, The Weyl calculus for Hermitian matrices, Proc. Amer. Math. Soc. 124 (1996), no. 1, 121–128. MR 1301032, https://doi.org/10.1090/S0002-9939-96-03143-7
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 Priložen 2 (1968), no. 2, 40–55 (Russian). MR 0236318
11. Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974). MR 0364552
12. Jean Jacques Loeb, Formule de Kirillov pour les groupes de Lie semi-simples compacts, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973–75), Springer, Berlin, 1975, pp. 230–256. Lecture Notes in Math., Vol. 497 (French). MR 0407204
13. Karl-Hermann Neeb, Holomorphy and convexity in Lie theory, De Gruyter Expositions in Mathematics, vol. 28, Walter de Gruyter & Co., Berlin, 2000. MR 1740617
14. Edward 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
15. L. Pukánszky, On the characters and the Plancherel formula of nilpotent groups, J. Functional Analysis 1 (1967), 255–280. MR 0228656
16. W. Rossmann, Kirillov's character formula for reductive Lie groups, Invent. Math., 48 (1978), 207-220. MR 0508985 (81g:22012)
17. Reyer Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), no. 1, 46–91. MR 1645052, https://doi.org/10.1006/aima.1998.1739
18. N. J. Wildberger, The moment map of a Lie group representation, Trans. Amer. Math. Soc. 330 (1992), no. 1, 257–268. MR 1040046, https://doi.org/10.1090/S0002-9947-1992-1040046-6