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Equivariant prequantization and admissible coadjoint orbits

Published online by Cambridge University Press:  24 October 2008

P. L. Robinson
P. L. Robinson
Affiliation:
Department of Mathematics, University of Florida, Gainesville FL32611, USA
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Extract

The orbit method has as its primary goal the construction and parametrization of the irreducible unitary representations of a (simply-connected) Lie group in terms of its coadjoint orbits. This goal was achieved with complete success for nilpotent groups by Kirillov[8] and for type I solvable groups by Auslander and Kostant[l] but is known to encounter difficulties when faced with more general groups. Geometric quantization can be viewed as an outgrowth of the orbit method aimed at providing a geometric passage from classical mechanics to quantum mechanics. Whereas the original geometric quantization scheme due to Kostant[9] and Souriau[14] enabled such a passage in a variety of situations, it too encounters difficulties in broader contexts.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 131 - 142
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Auslander, L. and Kostant, B.. Polarizations and unitary representations of solvable Lie groups. Invent. Math. 14 (1971), 255–354.CrossRefGoogle Scholar
[2]Blattner, R. J.. Quantization and Representation Theory. Amer. Math. Soc. Proc. Symp. Pure Math. 26 (1974), 147–165.CrossRefGoogle Scholar
[3]Duflo, M.. Sur les extensions des représentations irréductibles des groupes de Lie nilpotents. Ann. Sci. Ecole Norm. Sup. 5 (1972), 71–120.CrossRefGoogle Scholar
[4]Duflo, M.. Construction des représentations unitaires d'un groupe de Lie. Harmonic analysis and group representations, Liguori, Naples (1982), 129–221.Google Scholar
[5]Forger, M. and Hess, H.. Universal metaplectic structures and geometric quantization. Comm. Math. Phys. 64 (1979), 269–278.CrossRefGoogle Scholar
[6]Guillemin, V. and Sternberg, S.. Symplectic Techniques in Physics (Cambridge University Press, 1984).Google Scholar
[7]Hess, H.. On a geometric quantization scheme generalizing those of Kostant–Souriau and Czyż. Springer Led. Notes in Phys. 139 (1981), 1–36.CrossRefGoogle Scholar
[8]Kirillov, A.. Unitary representations of nilpotent Lie groups. Russian Math. Surveys 17 (1962), 53–104.CrossRefGoogle Scholar
[9]Kostant, B.. Quantization and Unitary Representations. Part I: Prequantization. Springer Lect. Notes in Math. 170 (1970), 87–208.CrossRefGoogle Scholar
[10]Kostant, B.. Symplectic Spinors. Symposia Mathematica 14 (1974), 139–152.Google Scholar
[11]Palais, R.. A global formulation of the Lie theory of transformation groups. Mem. Amer. Math. Soc. 22 (1957).Google Scholar
[12]Robinson, P. L.. Mpc structures, coadjoint orbits and admissible characters. Bull. London Math. Soc. 24 (1992), 289–292.CrossRefGoogle Scholar
[13]Robinson, P. L. and Rawnsley, J. H.. The metaplectic representation, Mpc structures and geometric quantization. Mem. Amer. Math. Soc. 410 (1989).Google Scholar
[14]Soubiau, J. M.. Quantification géométrique. Comm. Math. Phys. 1 (1966), 374–398.Google Scholar
[15]Varadarajan, V. S.. Lie Groups, Lie Algebras and their Representations (Prentice-Hall, 1974).Google Scholar