On the contraction of the discrete series of SU(1,1)
Annales de l'Institut Fourier, Volume 43 (1993) no. 2, p. 551-567

It is shown, using techniques inspired by the method of orbits, that each non-zero mass, positive energy representation of the Poincaré group 𝒫 1,1 =SO(1,1) s 2 can be obtained via contraction from the discrete series of representations of SU(1,1).

Nous montrons, en utilisant des idées provenant de la méthode des orbites, que toute représentation massive et d’énergie positive du groupe de Poincaré 𝒫 1,1 =SO(1,1) s 2 peut être obtenue par contraction de la série discrète de SU(1,1).

@article{AIF_1993__43_2_551_0,
     author = {Cishahayo, C. and Bi\`evre, S. De},
     title = {On the contraction of the discrete series of $SU(1,1)$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {2},
     year = {1993},
     pages = {551-567},
     doi = {10.5802/aif.1346},
     zbl = {0793.22005},
     mrnumber = {94e:22023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_2_551_0}
}
Cishahayo, C.; Bièvre, S. De. On the contraction of the discrete series of $SU(1,1)$. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 551-567. doi : 10.5802/aif.1346. http://www.numdam.org/item/AIF_1993__43_2_551_0/

[AAG] S. T. Ali, J. P. Antoine, and J.P. Gazeau, De Sitter to Poincaré contraction and relativistic coherent states, Ann. Inst. H. Poincaré, 52 (1990), 83-111. | Numdam | MR 91e:22027 | Zbl 0706.22018

[DBE] S. De Bièvre and A. El Gradechi, Quantum mechanics and coherent states on the Anti-de Sitter spacetime and their Poincaré contraction, Ann. Inst. H. Poincaré, 57 (1992), 403-428. | Numdam | MR 93m:81053 | Zbl 0770.53048

[D] A. H. Dooley, Contractions of Lie groups and applications to analysis, in : Topics in modern harmonic analysis, Vol.I, 483-515 (Instituto Nazionale di Alta Matematica Francesco SEVERI, Roma 1983). | MR 86e:22015 | Zbl 0551.22006

[DR1] A. H. Dooley and J. W. Rice, Contractions of rotation groups and their representations, Math. Proc. Camb. Phil. Soc., 94 (1983), 509-517. | MR 87a:22022 | Zbl 0532.22014

[DR2] A. H. Dooley and J. W. Rice, On contractions of semisimple Lie groups, Trans. Amer. Math. Soc., 289 (1985), 185-202. | MR 86g:22019 | Zbl 0546.22017

[E] A. El Gradechi, Théories classique et quantique sur l'espace-temps Anti-de Sitter et leurs limites à courbure nulle, Thèse de Doctorat, Université Paris 7, décembre 1991.

[EDB] A. El Gradechi and S. De Bièvre, Phase space quantum mechanics on the Anti-de Sitter spacetime and its Poincaré contraction, preprint 1992. | Zbl 0810.53070

[Fr] C. Fronsdal, Elementary particles in a curved space, Rev. Mod. Phys., 37 (1965), 221-224. | MR 31 #3193 | Zbl 0125.45305

[GH] J. P. Gazeau and V. Hussin, Poincaré Contraction of SU (1, 1) Fock-Bargmann Structure, J. of Physics A, Math. Gen., 25 (1992), 1549-1573. | MR 93k:81128 | Zbl 0824.22018

[Gi] R. Gilmore, Lie groups, Lie algebras and some of their Applications, Wiley, New York, 1974. | Zbl 0279.22001

[IW] E. Inönü and E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U. S., 39 (1953), 510-524. | MR 14,1061c | Zbl 0050.02601

[Ki] A. Kirillov, Eléments de la Théorie des Représentations, Editions Mir, Moscou, 1974.

[Ko] B. Kostant, Quantization and unitary representations, Lecture Notes in Math., 170, Springer Verlag, Berlin, 1970. | MR 45 #3638 | Zbl 0223.53028

[LM] P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, 1987. | MR 88c:58016 | Zbl 0643.53002

[Ma] G. W. Mackey, On the analogy between semisimple Lie groups and certain related semi-direct product groups, in : Lie Groups and Their Representations, I. M. Gelfand (Ed.), 339-364 (Akadémiai Kiadó, Budapest 1975). | MR 53 #13478 | Zbl 0324.22006

[MN] J. Mickelsson and J. Niederle, Contractions of representations of the Sitter groups, Commun. Math. Phys., 27 (1972), 167-180. | MR 46 #8611 | Zbl 0236.22021

[Pe] A. Perelomov, Generalized Coherent States and their Applications, Springer Verlag, Berlin, 1986. | MR 87m:22035 | Zbl 0605.22013

[R] P. Renouard, Variétés Symplectiques et Quantification, Thèse Orsay, 1969.

[Ra] J.H. Rawnsley, Representations of a semi-direct product by quantization, Math. Proc. Camb. Phil. Soc., 78 (1975), 345-350. | MR 52 #8341 | Zbl 0313.22014

[Sa] J. Saletan, Contraction of Lie groups, J. Math. Phys., 2 (1961), 1-21. | MR 23 #B259 | Zbl 0098.25804

[Wo] N.M.J. Woodhouse, Geometric Quantization, Clarendon Press, Oxford, 1980. | MR 84j:58058 | Zbl 0458.58003