A characterization of symmetric domains (Q862142)

The author proves that the commutativity of the Berezin transform with the Laplace-Beltrami operator characterizes the symmetric domains among all domains in \({\mathbb C}^n\) with a Kähler metric. Recall that Nomura proved in 2002 that this property characterizes the symmetric domains among the homogeneous ones. The proof is of differential geometric nature. It uses a relationship between the curvature tensor of the Kähler manifold \(\Omega \) and the derivatives of the reproducing kernel of the weighted Bergman space \(L^2_{\text{ hol}}(\Omega ,e^{-\Phi } d\mu )\), where \(\Phi \) is a Kähler potential and \(\mu \) the Riemannian measure associated with the Kähler metric.