, Volume 158, Issue 3, pp 285-305

Segal–Bargmann and Weyl transforms on compact Lie groups

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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).

Communicated by K.H. Gröchenig.
Research supported by Swedish Research Council (VR), and Swedish Foundation for International Cooperation in Research and Higher Education (STINT-PPP).