Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains (Q5945503)

This important paper by \textit{T. Nomura} is an extension of the author's work [RIMS Kokyuroku 1150, 72-80 (2000)]. The author begins the study of Berezin transforms on homogeneous Siegel domains without symmetry conditions. In the work of \textit{M. Engliš} [St. Petersbg. Math. J. 7, 633-647 (1996; Zbl 0859.58033); Algebra Anal. 7, 176-195 (1995; Zbl 0844.58084)], a result on Berezin transforms on domains in \(\mathbb{C}\) suggests that the commutativity of the Berezin transforms with the Laplace-Beltrami operator would stipulate the nature of the domain itself. The author making use of his result, namely Theorem 5.3 in the paper cited above, shows that on a homogeneous Siegel domain the Berezin transform commutes with the Laplace-Beltrami operator if and only if the domain is symmetric. The actual result which is the main result of this paper is slightly stronger and it states that the Berezin transforms \(B_\lambda\) commute with the Laplace-Beltrami operator \(L_\omega\) if and only if the domain \(D\) is symmetric and \(\omega\) is proportional to the Koszul form \(\beta\) on the derived algebra \([g,g]\).