Weyl quantization for semidirect products. (English) Zbl 1117.81087

Summary: Let \(G\) be the semidirect product \(V\rtimes K\) where \(K\) is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space \(V\). Let \(\mathcal O\) be a coadjoint orbit of \(G\) associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation \(\pi\) of \(G\). We consider the case when the corresponding little group \(K_0\) is a maximal compact subgroup of \(K\). We realize the representation \(\pi\) on a Hilbert space of functions on \(\mathbb R^n\) where \(n=\dim(K)-\dim(K+0)\). By dequantizing \(\pi\) we then construct a symplectomorphism between the orbit \(\mathcal O\) and the product \(\mathbb R^{2n}\times {\mathcal O}'\) where \({\mathcal O}'\) is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on \(\mathcal O\) which is adapted to the representation \(\pi\) in the sense of [B. Cahen, C. R. Acad. Sci., Paris, Sér. I 325, 803–806 (1997; Zbl 0883.22016)]. In particular we recover well-known results for the Poincaré group.


81S10 Geometry and quantization, symplectic methods
22E70 Applications of Lie groups to the sciences; explicit representations
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics


Zbl 0883.22016
Full Text: DOI


[1] Arnal, D.; Cahen, M.; Gutt, S., Representations of compact Lie groups and quantization by deformation, Acad. R. Belg. Bull. Cl. Sc. 3e série, LXXIV, 45, 123-141 (1988) · Zbl 0681.58016
[2] Arnal, D.; Cortet, J.-C., Nilpotent Fourier transform and applications, Lett. Math. Phys., 9, 25-34 (1985) · Zbl 0616.46041
[3] Arnal, D.; Cortet, J.-C.; Flato, M.; Sternheimer, D., Star-products: quantization and representations without operators, (Tirapegui, E., Field Theory, Quantization and Statistical Physics (1981), D. Reidel Publishing Company), 85-111
[4] Baguis, P., Semidirect products and the Pukansky condition, J. Geom. Phys., 25, 245-270 (1998) · Zbl 0932.22011
[5] B. Cahen, Star-représentations induites, Thèse, Université de Metz, 1992; B. Cahen, Star-représentations induites, Thèse, Université de Metz, 1992
[6] Cahen, B., Deformation program for principal series representations, Lett. Math. Phys., 36, 65-75 (1996) · Zbl 0843.22020
[7] Cahen, B., Quantification d’une orbite massive d’un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I, 325, 803-806 (1997) · Zbl 0883.22016
[8] Cahen, B., Quantification d’orbites coadjointes et théorie des contractions, J. Lie Theory, 11, 257-272 (2001) · Zbl 0973.22009
[9] Cahen, B., Contractions of \(SU(1, n)\) and \(SU(n + 1)\) via Berezin quantization, J. Anal. Math., 97, 83-101 (2005) · Zbl 1131.22005
[10] Cahen, M.; Gutt, S.; Rawnsley, J., Quantization on Kaehler manifolds I, Geometric interpretation of Berezin quantization, J. Geom. Phys., 7, 45-62 (1990) · Zbl 0719.53044
[11] Carinena, J.-F.; Garcia-Bondia, J.-M.; Varilly, J.-C., Relativistic quantum kinematics in the Moyal representation, J. Phys. A, 23, 901-933 (1990) · Zbl 0706.60108
[12] Cotton, P.; Dooley, A. H., Contraction of an adapted functional calculus, J. Lie Theory, 7, 147-164 (1997) · Zbl 0882.22015
[13] Folland, B., Harmonic Analysis in Phase Space (1989), Princeton University Press · Zbl 0682.43001
[14] Gotay, M., Obstructions to quantization, (J. Nonlinear Science, Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo) (2000), Springer: Springer New York), 271-316 · Zbl 1041.53507
[15] Guillemin, V.; Sternberg, S., Symplectic Techniques in Physics (1984), Cambridge University Press · Zbl 0576.58012
[16] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, vol. 34 (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0993.53002
[17] Helgason, S., Groups and Geometric Analysis, Mathematical Surveys and Monographs, vol. 83 (2000), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0965.43007
[18] Hormander, L., The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math., 32, 359-443 (1979) · Zbl 0388.47032
[19] Hormander, L., The Analysis of Linear Partial Differential Operators, vol. 3 (1985), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York, (Section 18.5) · Zbl 0612.35001
[20] Kirillov, A. A., Elements of the Theory of Representations, Grundlehren der mathematischen Wissenschaften, vol. 220 (1976), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0342.22001
[21] Kirillov, A. A., Merits and demerits of the orbit method, Bull. Amer. Math. Soc., 36, 4, 433-488 (1999) · Zbl 0940.22013
[22] Kostant, B., Quantization and unitary representations, (Modern Analysis and Applications. Modern Analysis and Applications, Lecture Notes in Mathematics, vol. 170 (1970), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York), 87-207 · Zbl 0223.53028
[23] Landsman, N. P., Strict quantization of coadjoint orbits, J. Math. Phys. 39, 12, 6372-6383 (1998) · Zbl 0986.81057
[24] Rawnsley, J. H., Representations of a semi direct product by quantization, Math. Proc. Camb. Phil. Soc., 78, 345-350 (1975) · Zbl 0313.22014
[25] Simms, D. J., Lie Groups and Quantum Mechanics, Lecture Notes in Mathematics, vol. 52 (1968), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0161.24002
[26] Taylor, M. E., Noncommutative Harmonic Analysis, Mathematical Surveys and Monographs, vol. 22 (1986), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0604.43001
[27] Voros, A., An algebra of pseudo differential operators and the asymptotics of quantum mechanics, J. Funct. Anal., 29, 104-132 (1978) · Zbl 0386.47031
[28] Wildberger, N. J., Convexity and unitary representations of a nilpotent Lie group, Invent. Math., 89, 281-292 (1989) · Zbl 0684.22005
[29] Wildberger, N. J., On the Fourier transform of a compact semi simple Lie group, J. Austral. Math. Soc. A, 56, 64-116 (1994) · Zbl 0842.22015
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.