Segal–Bargmann and Weyl transforms on compact Lie groups. (With Joachim Hilgert)

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as 3lafsson and 8rsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall\u27s 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 Kℂ. Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on Kℂ as operators on the Hilbert space L2(K)