Segal–Bargmann and Weyl transforms on compact Lie groups

摘要

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 . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on 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 Communicated by K.H. Gröchenig.