Illinois Journal of Mathematics

Contraction of compact semisimple lie groups via Berezin quantization

Benjamin Cahen

Full-text: Open access

Abstract

We establish contractions of the unitary irreducible representations of a compact semisimple Lie group to the unitary irreducible representations of a Heisenberg group by means of Berezin quantization.

Article information

Source
Illinois J. Math., Volume 53, Number 1 (2009), 265-288.

Dates
First available in Project Euclid: 22 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1264170850

Digital Object Identifier
doi:10.1215/ijm/1264170850

Mathematical Reviews number (MathSciNet)
MR2584946

Zentralblatt MATH identifier
1185.22008

Subjects
Primary: 22E46: Semisimple Lie groups and their representations 81R30: Coherent states [See also 22E45]; squeezed states [See also 81V80] 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]

Citation

Cahen, Benjamin. Contraction of compact semisimple lie groups via Berezin quantization. Illinois J. Math. 53 (2009), no. 1, 265--288. doi:10.1215/ijm/1264170850. https://projecteuclid.org/euclid.ijm/1264170850


Export citation

References

  • 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.
  • D. Bar-Moshe and M. S. Marinov, Realization of compact Lie algebras in Kähler manifolds, J. Phys. A: Math. Gen. 27 (1994), 6287–6298.
  • F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR Izv. 6 (1972), 1117–1151.
  • F. A. Berezin, Quantization, Math. USSR Izv. 8 (1974), 1109–1165.
  • F. A. Berezin, Quantization in complex symmetric domains, Math. USSR Izv. 9 (1975), 341–379.
  • B. Cahen, Deformation program for principal series representations, Lett. Math. Phys. 36 (1996), 65–75.
  • B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C.R. Acad. Sci. Paris t. 325 (1997), 803–806.
  • B. Cahen, Quantification d'orbites coadjointes et théorie des contractions, J. Lie Theory 11 (2001), 257–272.
  • B. Cahen, Contraction de SU(2) vers le groupe de Heisenberg et calcul de Berezin, Beiträge Algebra Geom. 44 (2003), 581–603.
  • 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.
  • B. Cahen, Contractions of $SU(1,n)$ and $SU(n+1)$ via Berezin quantization, J. Anal. Math. 97 (2005), 83–102.
  • B. Cahen, Berezin quantization on generalized flag manifolds, Preprint Univ. Metz (2008), to appear in Math. Scand.
  • B. Cahen, Multiplicities of compact Lie group representations via Berezin quantization, Math. Vesnik. 60 (2008), 295–309.
  • M. Cahen, S. Gutt and J. Rawnsley, Quantization on Kahler manifolds I: Geometric interpretation of Berezin quantization, J. Geom. Phys. 7 (1990), 45–62.
  • C. Cishahayo and S. de Bièvre, On the contraction of the discrete series of $SU(1,1)$, Ann. Inst. Fourier 43 (1993), 551–567.
  • L. Cohn, Analytic Theory of the Harish–Chandra C-function, Lecture Notes in Math., vol. 429, Springer, 1974.
  • P. Cotton and A.H. Dooley, Contraction of an adapted functional calculus, J. Lie Theory 7 (1997), 147–164.
  • 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.
  • A. H. Dooley and S. K. Gupta, The Contraction of $S^{2p-1}$ to $H^{p-1}$, Monatsh. Math. 128 (1999), 237–253.
  • 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.
  • A. H. Dooley and J. W. Rice, Contractions of rotation groups and their representations, Math. Proc. Camb. Phil. Soc. 94 (1983), 509–517.
  • A. H. Dooley and J. W. Rice, On contractions of semisimple Lie groups, Trans. Amer. Math. Soc. 289 (1985), 185–202.
  • B. Folland, Harmonic analysis in phase space, Princeton Univ. Press, 1989.
  • Harish-Chandra, Discrete series for semisimple Lie groups II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111.
  • S. Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics, vol. 34, Amer. Math. Soc., Providence, RI, 2001.
  • R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann. 290 (1991), 565–619.
  • 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.
  • E. Inönü, E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA 39 (1953), 510–524.
  • A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, vol. 64, Amer. Math. Soc., Providence, RI, 2004.
  • A. W. Knapp, Representation theory of semi simple groups. An overview based on examples, Princeton Math. Series, vol. 36, 1986.
  • K.-H. Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, vol. 28, Walter de Gruyter, Berlin, 2000.
  • J. Mickelsson and J. Niederle, Contractions of representations of de Sitter groups, Commun. Math. Phys. 27 (1972), 167–180.
  • F. Ricci, A Contraction of $SU(2)$ to the Heisenberg group, Monatsh. Math. 101 (1986), 211–225.
  • F. Ricci and R. L. Rubin, Transferring Fourier multipliers from $SU(2)$ to the Heisenberg group, Amer. J. Math. 108 (1986), 571–588.
  • V. S. Varadarajan, Lie groups, Lie algebras and their representations, Graduate Texts in Mathematics, vol. 102, Springer, New York, 1986.
  • N. R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, vol. 19, Marcel Dekker, New York, 1973.
  • N. J. Wildberger, On the Fourier transform of a compact semi simple Lie group, J. Austral. Math. Soc. A 56 (1994), 64–116.