Deformation program for principal series representations. (English) Zbl 0843.22020

The author begins with a parametrization of the principal series orbits of a semisimple Lie group \(G\) in terms of a cotangent bundle of a nilpotent Lie group and of a coadjoint orbit of a compact Lie group. Then he combines Berezin symbolic calculus on compact orbits and some generalization of Weyl correspondence to a cotangent bundle to obtain a symbolic calculus on the principal series orbits which is adapted to the description of the principal series representations. He shows that the \(*\)-product on principal series orbits associated to this symbolic calculus can be interpreted in terms of a Berezin product and on S. Gutt’s star product on the cotangent bundle of a nilpotent Lie group. Finally, the author applies his construction to define the adapted Fourier transform of \(G\) and he gives in the case where \(G\) has a unique class of Cartan subalgebras an elegant retranscription of the Plancherel formula.
Reviewer: J.Ludwig (Metz)


22E46 Semisimple Lie groups and their representations
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
58H15 Deformations of general structures on manifolds
Full Text: DOI


[1] Arnal, D., The * exponential, in Quantum Theories and Geometry, Kluwer Acad. Publ., Dordrecht, 1988.
[2] Arnal, D., Cahen, B., Cahen, M. and Gutt, S., Une classe d’orbites coadjoints qui sont symplectomorphes à un fibré cotangent, C.R. Acad. Sci. Paris, Serie I, 312, 127-130 (1991). · Zbl 0726.22012
[3] Arnal, D., Cahen, M., and Gutt, S., Representation of compact Lie groups and quantization by deformation, Acad. Royale Belg. Bull. Sci., LXXIV, 45, 3, 123-141 (1988). · Zbl 0681.58016
[4] Arnal, D., Cahen, M., and Gutt, S., * Exponential and holomorphic discrete series, Preprint U.L.B, 1988. · Zbl 0697.22010
[5] Arnal, D. and Cortet, J. C., Nilpotent fourier transform and applications, Lett. Math. Phys., 9, 25-34 (1985). · Zbl 0616.46041
[6] Arnal, D. and Cortet, J. C., Représentations * des groupes exponentiels, J. Funct. Anal. 92(1), 103-135 (1990). · Zbl 0726.22011
[7] Arnal, D., Cortet, J. C., and Ludwig, J., Preprint, Univ. Metz 1995.
[8] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Deformation theory and quantization, Ann. of Phys. 110, 61-151 (1978) · Zbl 0377.53024
[9] Cahen, B., Représentations étoiles induites, Thèse, Univ. Metz, 1995.
[10] Cahen, M., Gutt, S., and Rawnsley, J., Quantization of Kähler manifolds: geometric interpretation of Berezin’s quantization, J. Geom. Phys. 7(1), 45-62 (1990). · Zbl 0736.53056
[11] Cohn, L., Analytic Theory of the Harish Chandra C-Function, Lecture Notes in Math 429, Springer, Berlin, 1974. · Zbl 0342.33026
[12] Duflo, M., Fundamental-series representations of a semi simple Lie group, Funct. Anal. Appl. 4(2), 38-42 (1970). · Zbl 0254.22007
[13] Flato, M. et al., Simple facts about analytic vectors and integrability, Ann. Sci. Ecole Norm. Sup. 5(4), 423-434 (1972). · Zbl 0239.22019
[14] Fronsdal, C., Some ideas about quantization, Rep. Math. Phys. 15(1), 111-145 (1978). · Zbl 0418.58011
[15] Folland, B. Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989. · Zbl 0682.43001
[16] Gutt, S., An explicit * product on the cotangent bundle of a Lie group. Lett. Math. Phys. 7, 249-258 (1983). · Zbl 0522.58019
[17] Kirillov, A. A., Eléments de la théorie des représentations, Mir, Moscow, 1974.
[18] Knapp, A. W., Representation Theory of Semisimple Groups. An overview based on examples, Princeton Math. Series 36, Princeton Univ. Press, 1986. · Zbl 0604.22001
[19] Kostant, B., Quantization and Unitary Representations Lect. Notes Math. 170, Springer, Berlin, 1970. · Zbl 0223.53028
[20] Moore, C., Compactifications of symmetric spaces, Amer. J. Math. 86, 201-218 (1964). · Zbl 0156.03202
[21] Perelomov, A., Generalized Coherent States and Their Applications Springer, Berlin, 1986. · Zbl 0605.22013
[22] Voros, A., An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, J. Funct. Anal 29, 104-132 (1978). · Zbl 0386.47031
[23] Wallach, N. R., Harmonic Analysis on Homogeneous Spaces, Dekker, New York, 1973. · Zbl 0265.22022
[24] Warner, G., Harmonic Analysis on Semisimple Lie Groups I, II, Springer, Berlin, 1972. · Zbl 0265.22021
[25] Wildberger, N. J., Convexity and unitary representations of nilpotent Lie groups, Invent. Math. 98, 281-292 (1989). · Zbl 0684.22005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.