Monatshefte für Mathematik

, Volume 158, Issue 3, pp 285–305 | Cite as

Segal–Bargmann and Weyl transforms on compact Lie groups

Article

Abstract

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as Ólafsson and Ørsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and \({K_{\mathbb C}}\) . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on \({K_{\mathbb C}}\) as operators on the Hilbert space L 2(K).

Keywords

Segal–Bargmann transform Weyl transform Compact Lie group Hermite functions Reproducing kernel Toeplitz operator 

Mathematics Subject Classification (2000)

22E45 32A25 44A15 

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References

  1. 1.
    Berger C.A., Coburn L.A.: Heat flow and Berezin–Toeplitz estimates. Am. J. Math. 116, 563–590 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Duistermaat J.J., Kolk J.A.C.: Lie Groups. Springer, Berlin (2000)MATHGoogle Scholar
  3. 3.
    Hall B.C.: The Segal–Bargmann coherent state transform for compact Lie groups. J. Funct. Anal. 122, 103–151 (1994)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hall B.C.: The inverse Segal–Bargmann transform for compact Lie groups. J. Funct. Anal. 143(1), 98–116 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hall B.C.: Phase space bounds for quantum mechanics on a compact Lie group. Comm. Math. Phys. 184(1), 233–250 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hall B.C.: Harmonic analysis with respect to heat kernel measure. Bull. Am. Math. Soc. 38, 43–78 (2000)CrossRefGoogle Scholar
  7. 7.
    Hall B.C., Lewkeeratiyutkul W.: Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform. J. Funct. Anal. 217(1), 192–220 (2004)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hijab O.: Hermite functions on compact Lie groups I. J. Funct. Anal. 125, 480–492 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Neeb K.-H.: Holomorphy and Convexity in Lie Theory. de Gruyter, Berlin (2000)Google Scholar
  10. 10.
    Ólafsson G., Ørsted B. et al.: Generalizations of the Bargmann transform. In: Doebner, H.-D. (eds) Lie Theory and its Applications in Physics, World Scientific, Singapore (1996)Google Scholar
  11. 11.
    Ørsted B., Zhang G.: Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43, 551–582 (1994)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Robinson D.W.: Elliptic Operators and Lie Groups. Clarendon Press, Oxford (1991)MATHGoogle Scholar
  13. 13.
    Stenzel M.B.: The Segal–Bargmann transform on a symmetric space of conpact type. J. Funct. Anal. 165, 44–58 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III (1993)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany
  2. 2.Department of MathematicsChalmers University of Technology and Göteborg UniversityGöteborgSweden

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