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

Journal article, 2009

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 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).