*-products in the method of orbits for nilpotent groups. (English) Zbl 0599.22012

Authors’ abstract: ”On each orbit W of the coadjoint representation of any nilpotent (connected, simply connected) Lie group G, we construct *- products and associated Von Neumann algebras \({\mathfrak G}\). G acts canonically on \({\mathfrak G}\) by automorphisms. In the unique faithful, irreducible representation of \({\mathfrak G}\), this action is implemented by the unitary irreducible representations of G corresponding to W.”
Reviewer: S.Sankaran


22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
22E25 Nilpotent and solvable Lie groups
22D10 Unitary representations of locally compact groups
46L10 General theory of von Neumann algebras
Full Text: DOI


[1] Bayen, F.; Bayen, F., deformation theory and quantization, II, Annals of physics, Annals of physics, vol. III, 61, (1978) · Zbl 0377.53024
[2] Kostant, B.; Kostant, B., On certain unitary representations which arise from A quantization theory, Lect. notes in math., Lect. notes in math., 237, (1970) · Zbl 0229.20048
[3] Kirillov, A.A.; Kirillov, A.A., Unitary representations of nilpotent Lie groups, Ups. mat. nauk., Ups. mat. nauk., 57, (1962), (in Russian) · Zbl 0106.25001
[4] Arnal, D., (), 85
[5] Arnal, D.; Arnal, D., Covariance and geometrical invariance in ∗-quantization, J. math. phys., J. math. phys., 276, (1983) · Zbl 0515.22015
[6] Arnal, D.; Arnal, D., ∗-products and representations of nilpotent groups, Pac. J. of math., Pac. J. of math., 285, (1984) · Zbl 0561.58022
[7] Hansen, F., Quantum mechanics in phase space, Rep. math. phys., 20, (1984) · Zbl 0571.46046
[8] {\scJ.M. Maillard}, Preprint University of Dijon, 1982.
[9] Dixmier, J., LES C ∗-algebres at leurs representations, (1964), Gautier-Villars · Zbl 0152.32902
[10] Kastler, D.; Kastler, D., The C ∗-algebras of a free field I, Commun. math. phys., Commun. math. phys., 14, (1965) · Zbl 0137.45601
[11] Pukanski, L., Lecons sur LES representations des groupes, (1967), Dunod
[12] Lugo, V.; Lugo, V., An associative algebra of functions on the orbits of nilpotent groups, Lett. math. phys., Lett. math. phys., 509, (1981) · Zbl 0523.58018
[13] Goodman, R.; Goodman, R., Analytic and entire vectors for representation of Lie groups, Trans. amer. math. soc., Trans. amer. math. soc., 55, (1969) · Zbl 0189.14102
[14] Lion, G.; Lion, G., Integrales d’entrelacement sur des groupes de Lie nilpotents et indice de Maslov, Lect. notes. math., Lect. notes. math., 160, (1977) · Zbl 0391.22008
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.