×

Contraction of compact semisimple Lie groups via Berezin quantization. (English) Zbl 1185.22008

This paper establishes contractions of a compact semisimple Lie group. Let us recall that contractions of Lie groups have been introduced by Inonu and Wigner for physical purposes. Generally speaking, contraction theory is a way to relate the harmonic analysis on two Lie groups and enables one to recover as a byproduct the properties of special functions. In this paper, the author concentrates on the contractions of the unitary irreducible representations of a compact semisimple Lie group to the unitary irreducible representations of a Heisenberg group. The Berezin calculus is then introduced and related to the contraction of the Lie groups. Finally, the example of \(G= \text{SU}(p+q)\) is studied.

MSC:

22E46 Semisimple Lie groups and their representations
81R30 Coherent states
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] D. Arnal, M. Cahen and S. Gutt, Representations of compact Lie groups and quantization by deformation , Acad. R. Belg. Bull. Cl. Sc. 3e série LXXIV 45 (1988), 123–141. · Zbl 0681.58016
[2] D. Bar-Moshe and M. S. Marinov, Realization of compact Lie algebras in Kähler manifolds , J. Phys. A: Math. Gen. 27 (1994), 6287–6298. · Zbl 0843.58056 · doi:10.1088/0305-4470/27/18/035
[3] F. A. Berezin, Covariant and contravariant symbols of operators , Math. USSR Izv. 6 (1972), 1117–1151. · Zbl 0259.47004 · doi:10.1070/IM1972v006n05ABEH001913
[4] F. A. Berezin, Quantization , Math. USSR Izv. 8 (1974), 1109–1165. · Zbl 0312.53049 · doi:10.1070/IM1974v008n05ABEH002140
[5] F. A. Berezin, Quantization in complex symmetric domains , Math. USSR Izv. 9 (1975), 341–379. · Zbl 0324.53049 · doi:10.1070/IM1975v009n02ABEH001480
[6] B. Cahen, Deformation program for principal series representations , Lett. Math. Phys. 36 (1996), 65–75. · Zbl 0843.22020 · doi:10.1007/BF00403252
[7] B. Cahen, Quantification d’une orbite massive d’un groupe de Poincaré généralisé , C.R. Acad. Sci. Paris t. 325 (1997), 803–806. · Zbl 0883.22016 · doi:10.1016/S0764-4442(97)80063-8
[8] B. Cahen, Quantification d’orbites coadjointes et théorie des contractions , J. Lie Theory 11 (2001), 257–272. · Zbl 0973.22009
[9] B. Cahen, Contraction de SU(2) vers le groupe de Heisenberg et calcul de Berezin , Beiträge Algebra Geom. 44 (2003), 581–603. · Zbl 1032.22004
[10] B. Cahen, Contraction de \(SU(1,1)\) vers le groupe de Heisenberg , Mathematical works, Part XV, Université du Luxembourg, Luxembourg, Séminaire de Mathématique, 2004 pp. 19–43. · Zbl 1074.22005
[11] B. Cahen, Contractions of \(SU(1,n)\) and \(SU(n+1)\) via Berezin quantization , J. Anal. Math. 97 (2005), 83–102. · Zbl 1131.22005 · doi:10.1007/BF02807403
[12] B. Cahen, Berezin quantization on generalized flag manifolds , Preprint Univ. Metz (2008), to appear in Math. Scand. · Zbl 1183.22006
[13] B. Cahen, Multiplicities of compact Lie group representations via Berezin quantization , Math. Vesnik. 60 (2008), 295–309. · Zbl 1199.22016
[14] M. Cahen, S. Gutt and J. Rawnsley, Quantization on Kahler manifolds I: Geometric interpretation of Berezin quantization , J. Geom. Phys. 7 (1990), 45–62. · Zbl 0719.53044 · doi:10.1016/0393-0440(90)90019-Y
[15] C. Cishahayo and S. de Bièvre, On the contraction of the discrete series of \(SU(1,1)\) , Ann. Inst. Fourier 43 (1993), 551–567. · Zbl 0793.22005 · doi:10.5802/aif.1346
[16] L. Cohn, Analytic Theory of the Harish–Chandra C-function , Lecture Notes in Math., vol. 429, Springer, 1974. · Zbl 0342.33026
[17] P. Cotton and A.H. Dooley, Contraction of an adapted functional calculus , J. Lie Theory 7 (1997), 147–164. · Zbl 0882.22015
[18] A. H. Dooley, Contractions of Lie groups and applications to analysis , Topics in modern harmonic analysis, Proc. Semin., Torino and Milano 1982, vol. I, Ist. di Alta Mat, Rome, 1983, pp. 483–515. · Zbl 0551.22006
[19] A. H. Dooley and S. K. Gupta, The Contraction of \(S^2p-1\) to \(H^p-1\) , Monatsh. Math. 128 (1999), 237–253. · Zbl 0938.22007 · doi:10.1007/s006050050061
[20] A. H. Dooley and S. K. Gupta, Transferring Fourier multipliers from \(S^2p-1\) to \(H^p-1\) , Illinois J. Math. 46 (2002), 657–677. · Zbl 1018.22006
[21] A. H. Dooley and J. W. Rice, Contractions of rotation groups and their representations , Math. Proc. Camb. Phil. Soc. 94 (1983), 509–517. · Zbl 0532.22014 · doi:10.1017/S030500410000089X
[22] A. H. Dooley and J. W. Rice, On contractions of semisimple Lie groups , Trans. Amer. Math. Soc. 289 (1985), 185–202. JSTOR: · Zbl 0546.22017 · doi:10.2307/1999695
[23] B. Folland, Harmonic analysis in phase space , Princeton Univ. Press, 1989. · Zbl 0682.43001
[24] Harish-Chandra, Discrete series for semisimple Lie groups II. Explicit determination of the characters , Acta Math. 116 (1966), 1–111. · Zbl 0199.20102 · doi:10.1007/BF02392813
[25] S. Helgason, Differential geometry , Lie groups and symmetric spaces, Graduate Studies in Mathematics, vol. 34, Amer. Math. Soc., Providence, RI, 2001. · Zbl 0993.53002
[26] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions , Math. Ann. 290 (1991), 565–619. · Zbl 0733.20019 · doi:10.1007/BF01459261
[27] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains , Translations of Mathematical Monographs, vol. 6, Amer. Math. Soc., Providence, RI, 1963. · Zbl 0507.32025
[28] E. Inönü, E. P. Wigner, On the contraction of groups and their representations , Proc. Nat. Acad. Sci. USA 39 (1953), 510–524. · Zbl 0050.02601 · doi:10.1073/pnas.39.6.510
[29] A. A. Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics, vol. 64, Amer. Math. Soc., Providence, RI, 2004. · Zbl 1229.22003
[30] A. W. Knapp, Representation theory of semi simple groups. An overview based on examples , Princeton Math. Series, vol. 36, 1986. · Zbl 0604.22001
[31] K.-H. Neeb, Holomorphy and convexity in Lie theory , de Gruyter Expositions in Mathematics, vol. 28, Walter de Gruyter, Berlin, 2000. · Zbl 0936.22001
[32] J. Mickelsson and J. Niederle, Contractions of representations of de Sitter groups , Commun. Math. Phys. 27 (1972), 167–180. · Zbl 0236.22021 · doi:10.1007/BF01645690
[33] F. Ricci, A Contraction of \(SU(2)\) to the Heisenberg group , Monatsh. Math. 101 (1986), 211–225. · Zbl 0588.43007 · doi:10.1007/BF01301660
[34] F. Ricci and R. L. Rubin, Transferring Fourier multipliers from \(SU(2)\) to the Heisenberg group , Amer. J. Math. 108 (1986), 571–588. JSTOR: · Zbl 0613.43005 · doi:10.2307/2374655
[35] V. S. Varadarajan, Lie groups , Lie algebras and their representations, Graduate Texts in Mathematics, vol. 102, Springer, New York, 1986. · Zbl 0955.22500
[36] N. R. Wallach, Harmonic analysis on homogeneous spaces , Pure and Applied Mathematics, vol. 19, Marcel Dekker, New York, 1973. · Zbl 0265.22022
[37] N. J. Wildberger, On the Fourier transform of a compact semi simple Lie group , J. Austral. Math. Soc. A 56 (1994), 64–116. · Zbl 0842.22015 · doi:10.1017/S1446788700034741
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.