Monatshefte für Mathematik

, Volume 158, Issue 3, pp 285–305

Segal–Bargmann and Weyl transforms on compact Lie groups

Article

DOI: 10.1007/s00605-008-0080-0

Cite this article as:
Hilgert, J. & Zhang, G. Monatsh Math (2009) 158: 285. doi:10.1007/s00605-008-0080-0

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 L2(K).

Keywords

Segal–Bargmann transformWeyl transformCompact Lie groupHermite functionsReproducing kernelToeplitz operator

Mathematics Subject Classification (2000)

22E4532A2544A15

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